Formation of Photoelectron Holograms of Crystals Excited by Synchrotron Radiation V

DIFFRACTION AND SCATTERING OF X-RAY AND SYNCHROTRON RADIATION

Formation of Photoelectron Holograms of Crystals Excited by Synchrotron Radiation V. L. Nosik Center of Technological Innovations Crystal Growth, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia Received June 15, 2001

Abstract—The theoretical aspects of formation of photoelectron beams excited by the ultraviolet synchrotron radiation incident onto a crystal are considered. It is shown that the change in the energy of the monochromatized synchrotron radiation incident onto a hematite crystal results in the change of the number of reﬂections participating in the hologram formation. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION SPECIMEN The development of nanotechnology resulted in the The specimen was an orthorhombic hematite creation of various structural methods of visualizing α -Fe2O3 crystal with the unit-cell parameter a = 5.42 Å atomic structures which are based on the use of short- and α = 55.17° cut out in such a way that the hexagonal wavelength radiation (neutrons, electrons, X-ray and c-axis was normal to the entrance surface (Fig. 1). This synchrotron radiation, etc.). The term atomic hologra- crystal was selected because of the characteristic mag- phy coined by Gabor in 1948 [1] describes the method netic structure of the outer electron shells and its simple of the three-dimensional reconstruction of atomic structures due to positive interference of the strong ref- atomic structure. erence and weak object waves [2]. Later, the terms It is well known that the outer electrons of an iron atomic holography or the Fourier-transformation with- 6 2 atom (3d 4s ) obey the inequality E4s < E3d, which out a lens became widely used because of the well- explains the incomplete filling of the d-shell. It is inter- known optical analogy. esting to analyze the distributions of both collective However, the principle of image formation was 3d-electrons (magnetic structure) and 4s-electrons known long ago, e.g., as Kikuchi lines (in X-ray and (positions of individual atoms) in these crystals. Below, electron diffraction), X-ray standing waves in multi- we consider only the photoeffect of 4s-electrons. beam diffraction, Kossel lines, etc. [3, 4]. Lately, most of the published articles on this subject considered electron holography [5Ð7]. It was shown that, for high-energy electrons, the CowleyÐMoody calculations [8] yield images whose main features are the same as those of the experimental images [4]. However, p strong scattering of secondary electrons considerably z, c distorts the signal, which, in turn, hinders image processing. x At the same time, some schemes for studying photo- a1 and Auger electrons, elastically scattered X-ray radiation, fluorescence, etc. were created [9]. a y a Below, we consider the theoretical aspects of one of k 2 the new methods—positional-sensitive photoelectron spectroscopy—where excitation is caused by beams of soft synchrotron radiation. In fact, this method is a modern modification of photoelectron spectroscopy with angular resolution [10], which has been widely used for a long time to determine the band structure of Fig. 1. Geometry of scattering from an oriented hematite electrons. crystal.

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THEORY The coefficient B(p, t) determines the dependence of Within the framework of perturbation theory [11], the scattered radiation on time absorption of a photon with the potential vector A, the 2 ω expi()បω Ð E Ð p /2m Ð 1 wave vector Q, and the frequency (in the dipole Bpt(), = ------4s -. (8) approximation exp(iQa) ≈ 1, V = 1), បω 2 Ð E4s Ð p /2m 2π A = ------eexp()Ðiωt (1) When SR action on the atom is prolonged, the coef- ω ficient B(p, t) is simplified to the form results in electron transition (with a certain probability) π 2 from the initial state ψ localized at the atom R to one (), 2 2 tδp 2 4s i lim Bpt = ------------Ð k of the delocalized states, t → ∞ ប 2m (9) π ψ ()ψrRÐ , t = ()ψrRÐ + 0, 2 tm()δ()δ() i i 4s i i = ------ប ---- pz Ð kz + pz Ð kz (2) kz ψ 0 3 () (), i = ∫d piexp pr Ui B p t , and has the nonzero value only if the law of energy con- where the state of the photoelectron is described by the servation is obeyed. We used the following notation: integral over the plane waves exp(ipr), whereas the pulse p is measured in units ប. 2 បω 2 2 k ==+2E4s, kz mk Ð,pt (10) It should be emphasized that the calculation of the matrix element Ui of the transition where pt and pz are the components of the pulse vector lying in the plane of the crystal surface and normal to it, π 2 h ψ ()()∇ ()3 respectively. Ui = Ði ------ω ----∫ 4s rRÐ i e exp Ðipr d r (3) m Since the photoeffect is an inelastic process, no results in the appearance of the phase factor interference between the waves of the primary photoelectrons emitted by various atoms is possible, which () () Ui = epDpexp ipRi , (4) can readily be seen if one takes into account the arbi- where e is the polarization vector of the incident syn- trary phase factors before the wave functions of the core chrotron radiation (SR), and p, the wave vector of the electrons. The holographic effects can be explained if photoelectron. one takes into account the interference between the primary photoelectron radiation and the elastically scat- The coefficient D(p) is independent of the direction tered secondary photoelectron waves. The electron flux of the vector p because of the spherical symmetry of the ψ through the unit area J recorded by a two-dimensional wave function 4s. Thus, in the hydrogen-like approxi- detector (the surface normal n) is determined as mation we have ()φ ប ψ exp i 3/2 J = ------()ψ*()ψψn∇ Ð ()ψn∇ * . (11) 4s = ------Z im 44π (5) Using expression (2) and ignoring photoelectron ×ξ()ξ ξ2 2 1ξ3 3 exp Ð r 13Ð2r + r Ð --- r , scattering, we arrive at the following expression for 3 intensity, which has no term containing information on ξ the structure, where = Z/4aB, Z is the charge of the iron nucleus, the ប2 2 distances are measured in atomic units aB = /me = ប 0.529 Å, and φ is an arbitrary phase of the wave func- ∆ ∆ 3 () 2 (), 2τ () J0 S = S----∑∫d p np Ui Bpt i p , (12) tion. Differentiating the well-known expression m i 3 exp()Ð λr + ipr 4π τ ∫d r------= ------(6) where i(p) takes into account inelastic scattering and r p2 + λ2 absorption of photoelectrons prior to their detection, ∆ with respect to λ, we arrive at the cumbersome but rig- and S is the area of the detector pixel. orous expression for the function D(p), Now, take into account the scattering of the primary photoelectron wave propagating from the ith and scat- 1 Dp()∼ exp()iφ ------()ξ…+ . (7) tered by the jth atoms with the scattering field ϕ(r) p2 + ξ2 using perturbation theory (the kinematical or the Born approximation). The secondary wave generated by this Taking into account the high degree of monochro- atom is the convolution matization of the incident synchrotron radiation, we assume the pulse p and the coefficient D(p) to be con- ψ s () 3 ()ϕ()ψ0(), stant. ij R0 = Ð∫d rG R r j + r i r j + r t , (13)

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 FORMATION OF PHOTOELECTRON HOLOGRAMS 167 where R = R0 – rj – r. Expanding the wave function of we can represent the wave function of photoelectrons the primary photoelectrons, (2), and the Green’s func- scattered by all the atoms in the form tion into the Fourier series s 1 exp()ikR 3 exp()iqr ψ ()R = exp()ÐiH R GR()==Ð------d q------, i 0 ∑ jk 0 π ∫ 2 2 (20) 4 R q Ð k (14) jk 3 z 2 2 × (), ()(), k = p /2m, ∫d pUi Bptexp iptR0 F jk p R0 , we arrive at the following expression for the secondary where wave: 2 z 4π τ()p Ð q 3 () z z F jk p; R0 = ------∫dqz------ψ s () 3 (), d q 2 2 ()2 2 ij R0 = ∫d pUi Bpt∫------a qz + pt Ð H jk Ð k q2 Ð k2 (15) × ()()z (), × ()()()π () exp iqz Ð pz R0 f H jk pz Ð qz . exp iqR0 exp i pqÐ r j 4 f pqÐ , The function τ p q describes the effective obtained based on the definition of the structure fac- ( z – z) tor [12] decrease of the contributions from deeply located atomic planes f ()pqÐ = ∫d3rϕ()r exp()i()pqÐ r . (16) Z τ()p Ð q = ∑ exp Ð ip()Ð q Z Ð -----s , (21) Assuming that the atom is spherically symmetric, z z z z s L we obtain the atomic factor in the form s where L is the free path, which for electrons with the 2me2 ZfÐ ()s f ()s = ------p , (17) energy ranging within 10Ð1000 eV is almost constant ប2 s2 and equals several interatomic distances, and L ≈ 10 Å. Thus, only several upper atomic planes actively partic- where the term proportional to the number of protons, ipate in the signal formation. Z, corresponds to scattering from the nucleus and the With due regard for elastic scattering, the flux of term fp(s), to scattering by the electron shells of the electrons from the ith atom is the sum of the primary atoms. With an increase in s, the atomic-scattering fac- and the scattered waves [see (11) and (12)] tor rapidly decreases. Its absolute value equals F(0) = 4.74κ, F(h ) ≈ 3κ…, where κ = 2.393 × 10–8 cm [12]. 1 ប 3 2 2 2 The limitations of this approximation are considered in I ≈ ∆S----∑ d p()np U Bpt(), M()R ; p , (22) e m ∫ i 0 detail elsewhere [12]. The total wave function of the i secondary electrons is the sum over all the i atoms in s where the s atomic waves rj = Zsez – Ri , where ez is the unit vector normal to the crystal surface, Zs is the depth of () ()() s M R0; p = 1 + ∑ exp ÐiH jkR0 F jk p . the location of the sth plane, and Ri is the coordinate jk of the ith atom in the sth plane. With due regard for the law of energy conservation, The ideal surface structure of a hematite crystal is (9), integration of (22) over pz yields characterized by two vectors of the direct lattice, a1 and a (Fig. 1). Now, let the x-axis lie along the a vector. 2 1 ≈ π 2 (), 2 Then, the vectors of the two-dimensional direct and Ie 2 t∑∫dpt Ui M R0; pt kz . (23) reciprocal lattices (h1, 2) take the form i (),, ()α,,α a1 ==a 100, a2 a cos sin 0 , The above expression can be considered as the sum of three terms. The first term corresponds to the signal 2π 2π (18) h ==------()1,,Ð0cotα , h ------()01/,,sinα 0 . of the primary photoelectrons, (12), the second one, to 1 a 2 a the interference of the signals of the primary and scat- Using the two-dimensional variant of the well- tered photoelectrons, known summation rule, 0s ≈ π (), 2 Ie 4 t∑∫dpt Ui pt kz π2 i ()()s 4 δ() (24) ∑ exp i pt Ð qt Ri = ------2 ∑ pt Ð qt Ð H jk , (19) z i a jk, × ()(), Re∑ exp ÐiH jkR0 F jk pt kz; R0 ,  H jk = jh1 + kh2, jk

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Reflections in order of their appearance With an increase in the energy of the synchrotron- radiation quantum, the wavelength of a photoelectron Value No. H γ δε , eV increases, which results in the excitation of the first of γ, eV jk reflections with the minimum Hjk values. The threshold 1 H ()α 0.927 6.502 value of the energy providing the excitation of a given 11 21Ð cos reflection is determined by the following expression: 2 h1, h2 1 1 7.566 ()ប 2 α H jk 2 3 H21, H12 54Ð cos 1.649 20.573 δε ==បω Ð E ------≈ 7.566γ , (27) jk jk 4s 2m ()α 4 H1, Ð1 21+ cos 1.77 23.704 γ α π where = Hikasin /2 . The table indicates the calcu- 5 H22 221()Ð cosα 1.854 26.007 lated values of several reciprocal-lattice vectors and the corresponding energies of the synchrotron radiation. It should be indicated that electrons from other lev- and the third one (which is almost always very small els (e.g., 3d) are excited at much higher SR energies except for the region of the multibeam diffraction), to and, therefore, this contribution can be ignored. the interference of the secondary electrons. Thus, for an iron atom, the difference in the energies Below, we consider only term (24), which is respon- of the 3d and 4s levels in the hydrogen-like approxima- sible for the formation of the holographic image of the tion is about 500 eV (E4s = Ð574.6 and E3d = Ð1021.5 eV). crystal. It is convenient to represent this term in the At the same time, standard SR monochromators (e.g., form of a Fourier spectrum multilayer or segmented mirrors [13]) possess the energy resolution 0s ∼ ()() Ie Re∑ exp ÐiH jkR0 g jk k . (25) 3  E/δE ≈ 10 , jk sufficient for the formation of an SR beam with the The contributions of individual harmonics depend δ ≈ on the parameters of the two-dimensional crystal lattice energy width E 0.5 eV. and the photon energy (k) in a complicated way, The structure of the photoelectron beam emitted by the crystal is determined by the ratio between the coef- () , 2 , z g jk k = ∫ dpt Ui()pt kz F jk().pt kz; R0 (26) ficients gjk(k) before individual harmonics, so the main < pt k problem is reduced to their correct calculation. It should be indicated once more that the intensity of Ignoring the lattice relaxation near the surface (a = the holographic signal is proportional to the atomic const), we arrive at the absorption coefficient in the scattering factor (and not to its squared value as in con- form ventional scattering) because of the interference τ()≈ ()()()() Ð1 between the primary and the elastically scattered sec- pz Ð qz 1 Ð exp Ð ipz Ð qz + 1/L a . (28) ondary waves. This coefficient attains its maximum value, 1 C = ------(29) 4 1 Ð exp()Ða/L φ π at = (pz – qz)a = 2 m. The plots of the real and the imaginary parts of the function τ(φ) and also of its mod- 2 3 ulus are shown in Fig. 2. In the limiting case of a large 1 absorption length 2π τ()p Ð q ≈ ------∑δ()p Ð q Ð H , (30) z z a z z l 0 l

the integration over qz in Eq. (20) presents no difﬁcul- 2 ties π2 (), –2 0 (), z 8 f H jk 0 0 1 2 F jk p R0 = ------∑------. (31) φ 3 ()2 ()2 2 a l pz Ð Hl + pt Ð H jk Ð k With due regard for the law of energy conservation Fig. 2. (1) The real and (2) the imaginary parts of the func- 2 2 τ φ φ (p + p = k2), the latter equation starts converging tion ( ) and (3) its modulus as functions of = (pz – qz)a. z t

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 FORMATION OF PHOTOELECTRON HOLOGRAMS 169 with the approach to the Laue point of the given jkth 6 reflection (l = 0), (a) 1 ()L 2 ()L 2 pt = pt Ð H jk , (32) 2 because Eq. (2) was obtained within the framework of 4 perturbation theory (the Born approximation) invalid in 3 the vicinity of the Laue point. In this case, it would be expedient to use the two- beam Green’s function obtained from the system of equations for dynamical diffraction as the basis for con- 2 4 structing the solution. However, this approach would considerably complicate the calculation of the photoelectron signal, because the well-known Green’s functions can be used only within a narrow angular region 0 in the vicinity of the Bragg angle, whereas the integra- 1.5 tion should be made within much wider limits. Never- (b) theless, if one uses approximation (30), the contribution to the intensity from an arbitrary harmonic equals 0 () g jk k 1.0 π3 (33) 8 , 2 1 , = ------∫ dpt Ui()pt kz ------f (),H jk 0 a3 ()p Ð H 2 Ð p 2 p < k t jk t t 0.5 3 and the integrand acquires a pole as soon as a region in the reciprocal space is formed in which Laue condition (32) is 4 fulfilled. Such a region can be formed only if some photoelectrons have pulses exceeding half of the reciprocal 1 2 lattice vector, 2pt > H. 0 1 2 3 The analytical expressions for the coefficients ε g 0 (k) can be obtained in the approximation taking into jk Fig. 3. (a) Modulus of G and (b) the phase factor account the finite free path (28). As is shown in the jk () ⑀ Appendix, the first term of the asymptotic series for the H jk 2kHÐ jk as functions of = 2k/h for the coeffi- coefficient (27) in the far wave zone has the form cients before the first four harmonics.

2 ()∼ µ() ex ()()z g10 k k ------exp iH2kHÐ R0 , (34) Figure 3 shows the amplitude Gjk and the phase term ()z 5/2 R0 ⑀ H jk()2kHÐ jk as functions of = 2k/h for the first where four harmonics. The curve number corresponds to the reflection number in the table. A reflection can take an 3 3/2 28π k active part in the formation of the hologram only if ⑀ > µ()k = ÐDk()------τ()0 fH(), 0 ------. (35) 2 3/4 1/4 γ = 2H /h (see table). a 2H ()2kHÐ i i For further calculations, it is convenient to represent Figure 4 shows a number of holographic images for the contribution from an individual harmonic to a holo- different energies of an SR beam calculated by formu- gram as a product of the amplitude (G (k) = |g (k)|) and las (34) and (36). Each of the holograms formed upon jk jk the first one corresponds to the SR energy slightly the phase terms ⑀ γ exceeding the threshold energy i = 2k/h = i + 0.05, i = 1–4. 0s ∼ ()z Ie ∑ cos H jkR0 + H jk()2kHÐ jk R0 G jk().k (36) Upon the attainment of the threshold photon energy, jk the number of reflections increases in a jumpwise man- We should like to emphasize the importance of the ner (see table): H11 (Fig. 4a); H11, h1 (Fig. 4b); H11, h1, use of monochromatic synchrotron radiation. If it were H21, and H12 (Fig. 4c) and H11, h1, H21, H12, and H1, –1 not used the holographic signal would be suppressed by (Fig. 4d). It was assumed that the wave vector of the destructive interference of the waves excited by the synchrotron radiation and the surface normal photons with various energies k. formed an angle of 45° and that it was projected onto

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(a) (b) 2

–1 a

/ (c) (d) y 2

–10 1 2–1 0 1 2 x/a

Fig. 4. Calculated holographic images at different energies of the synchrotron radiation.

the surface along the direction of the vector a1, whereas When considering the experimental schemes, one z 5 has to take into account the ﬁnite dimensions of both the crystalÐdetector distance was equal to R0 = 10 a. specimen and incident beam. As a result, the diffracted The polarization vector of the synchrotron radiation is photoelectron beams corresponding to different diffrac- 1 tion vectors diverge in the space, and the interference e = ------(1, 0, –1). The polarization of the synchrotron 2 pattern is formed only in the region of their intersection. However, considering only the problem of the pre- radiation prevents the harmonic corresponding to vec- cise determination of the lattice parameter along a cer- h tor 2 from participating in the hologram formation. tain crystallographic direction, one can examine only Usually, the SR used in the experiments is both the “striped” hologram formed from only one reﬂection monochromatized and linearly polarized. Rotating the (Fig. 4a). specimen around the beam, it is possible to vary the Moreover, the record of complicated holograms is contribution of each harmonic H because of the change rather time-consuming, even if powerful SR sources of the factor (eH)2/H2. and position-sensitive detectors are used.

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CONCLUSION APPENDIX In terms of physics, all the holographic methods are CALCULATION OF CONTRIBUTIONS FROM INDIVIDUAL HARMONICS based on the well-known phenomenon of interference of the primary wave and the waves elastically scattered In fact, this Appendix deals with the calculation of by ordered atoms in a crystal. In practice, atomic holog- integral (27). As an example, consider the coefficient raphy can be used for extracting structural information before the harmonic with H10. in the case of relatively weak scattering. z The integrand in the expression for Fjk(pt, kz; R0 ) Most probably, photoelectron beams cannot com- (), z pete with the modern technologies used in optics and F10 pt kz; R0 (A.1) fiber optics to transfer information. However, we 2 4π τ()k Ð q f ()H, k Ð q z believe that the use of photoelectrons in scientific = ------dq ------z z z z exp()iq()Ð k R 2 ∫ z 2 ()2 2 z z 0 experiments is promising because of the extremely high a qz + pt Ð H Ð k sensitivity of low-energy electron beams transmitted 12, ±κ κ by crystalline objects to their structures. Moreover, in includes with the pole points qz = , = recent decades, we have seen the rapid development of 2 t 2 efficient methods for electron beam control (enlarge- k Ð ()p Ð H formed due to the singularities of the ment, focusing, etc.) and the transformation of elec- 34, ± Green’s function and qz = kz H associated with the tron images into corresponding analogous and digital form of the atomic scattering factor. Formally, the last optical images, which gives us hope for the further two points arise because of scattering from a single successful development of combined holographic atom accompanying the U-processes and thus have no methods. physical meaning. The small escape depth of photoelectrons allows Also, one has necessarily to take into account the τ one to record a signal from the subsurface region, contribution from the function . One can readily see that one of the best approximations of this function in where the lattice is often rather distorted by relaxation ≈ and surface reconstruction. While the method of X-ray the vicinity of the maximum (qz kz) is standing waves allows the determination of distances τ() 1 with an accuracy of 0.1% of the interplanar spacing qz Ð kz = ------()π () - 12Ð exp iqz Ð kz a Ð 0.5 along the surface normal, the photoelectron hologram (A.2) allows the determination of the lattice periodicity in the ≈τ 0.15i m Ð ------()-, surface plane. aqz Ð kz Ð i0.08/a τ Photoelectron holography can be used not only in where m = 0.622 is the minimum value of the initial studies of crystal structures, but also in studies of function. In this approximation, the function τ acquires objects with dimensions not exceeding several nanom- a pole, q 5 = k + i0.08/a. eters. If one places any noncrystalline object or an z z object possessing lattice parameters different from Integration upon the closure of the integration con- 1 5 those of the crystal, the hologram preserves its peri- tour with respect to qz around the poles qz and qz odic structure, but the region coated by this object is yields considerably shadowed. This can be efficiently used for visualizing the structures of complex objects such π3  (), z 8 i ()()κ z F10 pt kz; R0 = ------exp Ðikz Ð R0 --- as Langmuir–Blodgett films and films of other organic 2κ  materials. a f ()H, κ Ð q 0.15i z Thus, the main advantages of photoelectron holog- ×τ()k Ð κ ------z- Ð ------exp()Ð0.08R /a z 2κ a 0 raphy in the case of photoelectron excitation by SR (A.3) beams are: f ()H, i0.08/a  × ------. —the possibility of analyzing the distribution of ()k + i0.08/a 2 + ()p Ð H 2 Ð k2  electrons of a certain level because of high coherency z t and monochromaticity of highly polarized SR beams; Obviously, the exponential factor before the second —the dependence of the number of reflections term in (A.3) levels its contribution, because the detec- forming holograms on the energy of an SR beam; torÐcrystal distance usually exceeds the interatomic distance by several orders of magnitude. —the possibility of determining the structure of Below, we consider only the contribution from the crystal surfaces. pole point of the Green’s function. Since the value k is

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 172 NOSIK fixed, some factors in Eq. (26) can be taken outside the where integral ≡ (), () b* bpx p* = H 2 px Ð H , k p* (A.8) () ()2 ()Γz (), Γ 2 2 Γ 2 g10 k = Dk ∫dpx ∫ dpy exp ÐibR0 px py , 1 ==ex px / p*, 2 ey p*. 0 Ðp* (A.4) 3 One can readily see that the integration of the 4π i 2 Γ Γ()p , p =------()e p ++e p e k τ()b f ()H, κ Ð q expression containing 2 with respect to dpx yields x y 2 x x y y z z z , Γ a κ zero. At the same time, the integral of 1 diverges with an approach of px to k. Using the Erdélyi lemma [14], 2 2 κ this integral with the diverging integrand can be evalu- where p* = k Ð px , b = kz – . It should be indicated that, formally, the limits of integration with respect ated by the expression to px, y range from minus to plus infinity, but, in fact, a ∞ k + β Ð------they are limited to a circle of radius k beyond which the β Ð 1 () ()λ α ∼ λ α κ ∫x fxexp i x dx ∑ ak (A.9) function kz – becomes complex. 0 k = 0 The expression at λ ∞, where κ bk= z Ð ()k () β π()β (A.5) f 0 Γk + i k + 2 2 2 2 ()2 2 ak = ------------exp------. (A.10) = k Ð px Ð py Ð k Ð px Ð H Ð py k!α α 2α Using an indefinite integral, we substitute the can go to zero at the Laue point (2px* = H) if 2k > H. expression The first and second derivatives of b with respect to px 1 z z z also go to zero at the same point, which does not allow ------exp()ib*R = iibexp()*R dR the use of the saddle-point method to calculate an inte- b* 0 ∫ 0 0 z ∞ gral with a high value of the phase factor (R0 ). into expression (A.7). However, a high value of the phase factor allows one to The major contribution to the integral with respect represent the integrals as expansions in the degrees to p z –N x of ()R0 , 3 2 p* 216π i ex z g ()k = ÐiD() k ------()Γ() 10 2 z 2 Ip= d y exp ÐibR0 py () ∫ a R0 Ðp* (A.11) k 2 N k z px z Ð1k Ð Ð1 d Γ()p (A.6) × dR dp ------τ()b* fHb(), * exp()ib*R () y ∫ 0 ∫ x 0 = ∑ iR0 ------()------() p* b' py dpy b' py 0 k = 0 β α p* comes from the region px ~ k, where = 1/2 and = 1, × ()z ()ÐN exp ÐibR0 + oR0 . Ðp* 3/2 ∼ k τ , ()z ()λ Here, we limit our consideration to the first nonvan- f ------()0 f ()H 0 exp iH()2kHÐ R0 exp i px , Ð2 2 ishing term of this series (~R0 ), although, in order to (A.12) z increase the accuracy, one has to take into account the 2 HR λ = ------0 . following terms [14]. The calculations with the use of 2kHÐ the first term of the expansion yield zero because of 1 Thus, the first term of expansion (A.9) yields the fol- lim------= 0 lowing expression for the coefficient before the p → p*b'()p y y harmonic: and, therefore, one has to take into account the second 2 3 term as well ex 28π 3/2 g ()k ∼ Ð------Dk()------k τ()0 f ()H, 0 10 z 2 2 3 () a 1 32π i R0 I ≈ ------τ()b* fHb()Γ, * ()+ Γ (A.13) z 2 2 1 2 1/4 ()R p* a b* (A.7)  0 × 2kHÐ ()z, ------LR0 2kHÐ , × ()z 16H exp ib*R0 ,

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 FORMATION OF PHOTOELECTRON HOLOGRAMS 173 where One should pay attention to an unusual dependence of the signal intensity on the detectorÐobject distance, z z 1 z (), ()() z LR0 2kHÐ = dR0 ------exp iH2kHÐ R0 . –5/2 ∫ z ()R0 . For comparison, for a point source the signal R0 intensity is I ~ (R z )–2. At 2k – H ≠ 0, let the infinite integral determining L can 0 z –k – 1/2 be represented as an expansion in degrees of ()R0 . The first term of this expansion yields REFERENCES 1. D. Gabor, Nature 161, 777 (1948). ()z, LR0 2kHÐ 2. E. Wolf, Opt. Commun. 1, 153 (1969). ∼ 1 ()()z (A.14) 3. J. M. Cowley, Diffraction Physics (Elsevier, New York, ------exp iH2kHÐ R0 . 1975; Mir, Moscow, 1979). ()z H 2kHÐ R0 4. A. L. Danishevskiœ and F. N. Chukhovskiœ, Kristal- Finally, we have lografiya 27, 668 (1982) [Sov. Phys. Crystallogr. 27, 401 (1982)]. 2 ()∼ µ() ex ()()z 5. J. J. Barton, Phys. Rev. Lett. 61, 1356 (1988). g10 k k ------exp iH2kHÐ R0 , (A.15) ()z 5/2 6. D. K. Saldin, G. R. Harp, B. L. Chen, and B. P. Tonner, R0 Phys. Rev. B 44, 2480 (1991). where 7. L. J. Terminello, J. J. Barton, and D. A. Lapiano-Smith, Phys. Rev. Lett. 70, 599 (1993). 3 3/2 28π k µ()k = ÐDk()------τ()0 f ()H, 0 ------. 8. J. M. Cowley and A. F. Moodie, Acta Crystallogr. 10, a2 2H3/4()2kHÐ 1/4 609 (1957). (A.16) 9. B. Adams, D. V. Novikov, T. Hiort, and G. Materlik, Phys. Rev. B 57, 7526 (1998). Obviously, if the Laue condition (2k > H) is not fulfilled, q (k) decreases with an increase in the crystalÐ 10. B. Feuerbacher and B. Fitton, in Electron Spectroscopy 10 for Surface Analysis, Ed. by H. Ibach (Springer-Verlag, z detector distance R0 . If the Laue condition is met at Berlin, 1977), p. 151. least in a part of the space, the coefficient decreases 11. S. Flugge, Practical Quantum Mechanics (Springer-Ver- proportionally to lag, Berlin, 1971; Mir, Moscow, 1974). Ð1/4 12. B. K. Vainshtein, Structure Analysis by Electron Diffrac- ()2kHÐ tion (Akad. Nauk SSSR, Moscow, 1956; Pergamon, with an increase in the photon energy. Oxford, 1964). Formally, the divergence of the result obtained at 13. A. V. Vinogradov, Mirror X-ray Optics (Mashinostroe- 2k = H is associated with the form of the Green’s func- nie, Leningrad, 1989). tion used. 14. M. V. Fedoryuk, Asymptotics: Integrals and Series The contribution from the given reflection is maxi- (Nauka, Moscow, 1987). mal, if the polarization vector of the incident radiation is parallel to the reciprocal-lattice vector (ex = (eH)/H = 1). Translated by L. Man

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

DIFFRACTION AND SCATTERING OF X-RAY AND SYNCHROTRON RADIATION

X-ray Focusing by Bent Crystals in Bragg Backscattering T. Tchen Lomonosov Academy of Fine Chemical Technology, pr. Vernadskogo 86, Moscow, 117571 Russia e-mail: [email protected], [email protected] Received July 27, 2001

Abstract—Focusing of a spherical X-ray wave in Bragg backscattering from weakly and strongly bent crystals is considered theoretically. The analytical formula describing the dimensions of the diffraction backscattering region is obtained. It is shown that, along with the well-known Johann scheme, the use of the backscattering scheme allows one to increase the aperture ratio of the crystal optics by two to three orders of magnitude. The spectral characteristics of bent crystals providing diffraction backscattering (θ = π/2) are discussed. It is shown that the spectral resolution can attain a value of the order of 10Ð11. © 2002 MAIK “Nauka/Interperiodica”.

Backscattering of the X-ray radiation occurring at ittal plane can exceed the divergence in the meridional θ ≅ π the diffraction angle /2 has some specific features plane by two to three orders of magnitude because yeff ~ that distinguish it from conventional diffraction at θ < (∆θ)1/2 ~ |χ |1/2. For backscattering, the substitution of π hr /2. The width of the dynamic curve of the Bragg cosθ ~ θ into Eq. (1) yields y ~ ∆θ ≤ |χ |1/2. An addi- ∆θ |χ |1/2 –2 –3 eff hr reflection equals 2 = 2 hr ~ 10 –10 [1Ð4], tional increase in the aperture ratio can be attained by which allows an increase in the aperture ratio of an opti- biaxial bending of the crystal with the curvature radii cal device by two to three orders of magnitude. The use R ~ R . of bent crystals focusing an X-ray beam in one or two x y dimensions allows an increase in the aperture ratio of Below, we consider one-dimensional focusing of a several orders of magnitude. Moreover, as was shown spherically divergent X-ray beam in its backscattering in [5], backscattering also provides the minimization of by weakly or strongly bent crystals. The crystals are the geometrical aberration of the diffracted beam. The assumed to be bent in the meridional plane. The results well-known Johann focusing scheme [6] in which both obtained can readily be extended to biaxial bending by radiation source and detector are located on the Row- using the theory of two-dimensional focusing devel- land circle is widely used also because of its high aper- oped in our earlier work [12]. The multibeam effects ture ratio. Johann focusing was repeatedly studied [7Ð accompanying backscattering [13] are ignored. 11]. The best possibilities for obtaining a high aperture ratio are provided by the Johann and the backscattering REGION OF DIFFRACTION REFLECTION schemes. Theoretically, the upper limit of the aperture IN BACKSCATTERING ratio in both schemes is attained with the aid of biaxially bent crystals. The results obtained in [5] show that In this section, the Bragg diffraction reflection of a the aperture ratio of a biaxially bent X-ray lens is pro- spherical X-ray wave by a bent crystal is considered in portional to the geometrical-optical approach of radiation propagation. Earlier [5], we considered in a similar way the Ω θ 2 = 4sin xeff yeff/L0 , Bragg diffraction of an X-ray beam reflected from a biaxially bent crystal [5] at the diffraction angles θ ≠ π/2. where Let a spherical X-ray wave from a point source S be ∆θ θ xeff = L0 / sin Ð,L0/Rx incident onto a cylindrically bent crystal at the Bragg angle θ. Upon diffraction from the crystal, the X-ray 1/2 (1) y = L 2∆θcos θ/()sinθ Ð L ()1 + sin2 θ /R , radiation is focused into a line normal to the diffraction eff 0 0 y scattering plane intersecting this plane at the point S'. ∆θ is the halfwidth of the Bragg reflection curve, L0 is The scheme of focusing of an X-ray beam reflected in the distance between the source of a spherical X-ray the backward direction by uniaxially bent crystal is wave and the crystal, and Rx and Ry are the curvature shown in Fig. 1. radii of the crystal in the meridional and the sagittal For the sake of simplicity, we limit our consider- planes, respectively. The formula for xeff was obtained ation to the symmetric diffraction. At the Bragg angles ≠ θ θ ≠ π under the assumption that L0 Rxsin . It is seen from /2, the size xeff of the region on the crystal surface Eq. (1) that at θ ≠ π/2, the beam divergence in the sag- where the diffracted beam is within the boundary of the

X-RAY FOCUSING BY BENT CRYSTALS 175 region of total reﬂection is the solution of the quadratic S' equation ∆θ | θ = Ð xeff/Rx + xeff sin /L0 2 ()2 θ θ Ð xeff 2sin Ð 1 /2Rx L0 cos (2) S 2 θθ2 2 θ 2 θ Ð xeff sin cos /L0 + xeff sin /2Rx cos Z 2 2 θ()θ 2 θ| + xeff sin sin Ð L0/Rx /2L0 cos . ≠ θ If L0 Rxsin , the spherical aberration can be ignored and the terms proportional to x2 in Eq. (2) can be rejected. Then Eq. (2) yields an equation linear with respect to xeff which, in turn, yields Eq. (1). For the Johann scheme, Eq. (2) yields

1/2 X x , = R ()2∆θtan θ , eff Johann x (3) θ θ ≠ π L0 = Rx sin , /2. ∆θ |χ | θ Here, = hr /sin2 is the angular halfwidth of the region of total reflection (the Bragg reflection curve), χ hr is the real part of the Fourier-component of X-ray polarizability, and Rx is the curvature radius of the crys- Fig. 1. Principal scheme for the focusing of a spherical tal in the meridional plane (the diffraction reflection back-scattered wave by a cylindrically bent crystal. S is a plane). point source and S' is its image. ∆θ In backscattering, xeff and are related as

2 2 ∆θπ = x [()1/R ++()1/L ()2/R L fourth order with respect to x. Moreover, we also /2 eff x 0 x 0 Ӷ (4) assumed that xeff /Rx 1. 2 ()]2 2 3 4 4 1/2 + xeff 1/2Rx L0 Ð1/Rx L0 Ð 3/4L0 Ð 3/4Rx . ӷ If a plane wave is incident onto a crystal (L0 Rx), FOCUSING OF SPHERICAL AND PLANE WAVES Eq. (4) yields the angular halfwidth of the reflection IN BACKSCATTERING FROM A WEAKLY curve in the form BENT CRYSTAL ∆θ ()2 2 1/2 Consider a crystal weakly bent in the meridional π/2 = xeff 13Ð xeff /4Rx /Rx. (5) plane so that the dimensionless bending parameter Ӷ |ν| ӷ 1 (see, e.g., [12]). In the dynamical focusing of a It follows from (5) that if xeff Rx, then the size of the diffraction-reflection region in the reflection of the spherical X-ray wave by a weakly bent crystal, the plane wave in the backward direction (θ ≅ π/2) is intensity in the focus is distributed by the law 2 1/2 2 Ð1 Ð1 x ≈ R ∆θπ ≅ R χ . (6) I ()ξ = E ()ξ ∼ J ()π∆ξ ξ /π∆ξ ξ , eff x /2 x hr h p h p 1 p p p p (9) ≠ θ If a source of the spherical wave is located at a distance L0 Rx sin , L0 = Rx from the crystal, Eq. (4) yields ξ where p is the coordinate of the point of the source 2 2 image in the transverse direction, J is the first-order ∆θπ = 2x ()1 Ð x /2R /R . (7) 1 /2 eff eff x x Bessel function of the real argument, and ∆ξ is the dif- Ӷ p Taking into account that xeff /Rx 1, we obtain from fraction broadening of the focus. Eq. (7) Equation (9) was obtained under the assumption ≈ χ 1/2 that the phase of an incident spherical wave is expanded xeff Rx hr /2. (8) into a series in powers of x/L0 (up to the second-order Expressions (3), (6), and (8) show than the Johann and terms). The use of the parabolic expansion of the wave ≠ θ the backscattering schemes provide the gain in the phase at L0 Rxsin signifies that the spherical aberra- aperture ratio exceeding by two to three orders of mag- tion is ignored. However, in the Johann scheme, one has nitude the gain provided by all the other focusing also to take into account the terms ~x3, which results in schemes. It should also be indicated that, when deriving the redistribution of the intensity in the form of the Eq. (4), we took into account only the terms up to the squared modulus of the Airy function [14].

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 176 TCHEN π/4 π/2 θ ANGULAR SHIFT OF THE BRAGG MAXIMUM IN BACKSCATTERING FROM A BENT CRYSTAL –1 It is well known [15] that, because of refraction, the Bragg angles in a vacuum and in a crystal are different. χ θ The shift of the Bragg angle equals or/sin2 . In the δ Bragg reflection from a bent crystal, crystal bending log gives rise to an additional shift of the maximum of the –4 reflection curve [11]. This shift is essential in two limiting instances: at θ 0 and θ π/2 (Fig. 2). In the case of the backscattering we are interested in, the Fig. 2. Logarithm of the shift δ as a function of the diffrac- angular shift equals θ λ θ χ tion angle . Si (111), = 1.55 Å ( = 14.31°), or = − × –5 χ × –6 –4 δ ≅ ()χ χ 1/2 1.533 10 , hr = –7.992 10 , T/Rz = 10 . T/Rz Ð or /2 hr . (13) χ χ At the same values of T, or, and hr as in [11], we According to [12], the diffraction broadening of the obtain from formula (13) that δ ≤ 10–2–10–1; in other focus in Eq. (9) is words, the shift exceeds the width of the reflection ∆ξ θ Λθ curve. To decrease the shift of the Bragg angle, one has p = 1 Ð Lh/Rx sin cos , (10) to use thin crystals with a thickness satisfying the con- ≠ θ ≈ χ Lh Rx sin , ditions T Rz or, whence it follows that, for bent crystals, the shift can be less than for a plane crystal. Λ where Lh is the crystalÐimage distance, = λ θ |C||χ | is the extinction length in the symmet- It is seen from (13) that this shift becomes essential sin /( hr ) ≥ –4 λ if T/Rz 10 and its order of magnitude considerably ric case, is the wavelength of the incident radiation, χ and C = ±1 (for θ = π/2) is the polarization factor. differs from that of or or else for thin crystals where T/R Ӷ χ . In the backscattering of a spherical wave, the dif- z or ∆ξ ≤ | fraction broadening of the focus is p 1 – |Λ |χ | |∆λ λ| 1/2 ∆λ λ Lh /Rx ( hr + / ) , where / takes into FOCUSING SPECTROMETER account the nonchromaticity of the incident wave. IN BACKSCATTERING FROM A WEAKLY BENT CRYSTAL The sourceÐcrystal, L0, and the crystalÐimage, Lh, distances for a point source satisfy the modified equa- Consider the spectral properties of a weakly bent tion of an X-ray lens [11], crystal at θ = π/2. The linear dispersion Dξ is

1/L0 + 1/Lh Dξ = dξ /dλ ≅ ()L Ð L dθ/dλ. (11) p 0 h (14) = 2[]() 1+ χ /2sin2 θ /1()+ T/2R /R sinθ, or z x Using (10), we obtain for the spectral resolution where T is the crystal thickness and R is determined by z λ λ∆ξ θ ()θ the components of the inverse elastic-modulus tensor d / = p cos / L0 Ð Lh sin [11]. The corrections in the right-hand side of Eq. (11) ≅ 1 Ð L /R sinθ λ∆θ()2/ C χ ()θL Ð L sin take into account the radiation refracted by the crystalÐ h x hr 0 h (15) vacuum interface and the change in the interplanar ≅ 1 Ð L /R λ/ CL()Ð L , spacing in the crystal depth. These corrections give, in h x 0 h –5 –4 ≠ fact, very small (~10 –10 ) contributions and, there- L0 Lh. fore, can be ignored in most of the cases of practical interest. Diffraction reﬂection is coherent for the sources with ϕ ≈ λ the angular dimensions a/L0 < /xeff. Therefore, If a plane wave is reﬂected from a crystal, the dif- ≤ ∆ξ Λ θ Eq. (15) is valid for the sources of the size a L0/xeff. fraction broadening of the focus equals p = cos /2. λ ∆ξ θ ≈ π θ∆θ ≈ ∆θ ≤ If a > L0 /xeff, the quantity p in Eq. (15) should be In backscattering, cos cos( /2) + sin ∆ξ λ |χ | |∆λ λ| 1/2 replaced by p + L0 /xeff. ( hr + / ) , i.e., the diffraction broadening of the Ӷ θ focus equals At Lh Rxsin , the spectral resolution described by Eq. (15) is dependent on the Bragg angle θ and ∆ξ ≤ λχ()∆λ λ 1/2 χ p hr + / /2 C hr . (12) dλ(θ)/λ ~ f(θ) = (1 + cos2θ)/2sinθ and decreases with θ |∆λ λ| Ӷ an increase in (Fig. 3). Under the assumption that If an incident wave is highly monochromatic ( / L ӷ R , we arrive at the following theoretical estimate |χ | ∆ξ ≤ λ | ||χ |1/2 0 x hr ), Eq. (12) yields p /2 C hr . In the reflec- of the spectral resolution for a plane incident wave: tion of the MoKα radiation from the Si(444) plane, the ∆ξ ≈ µ λ λλ∼ focus size equals p 0.037 m. d / /2 CL0. (16)

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 X-RAY FOCUSING BY BENT CRYSTALS 177

The theoretical limit of the spectral resolution (16) in +∞ backscattering of the CuKα radiation (λ = 1.54 Å) from () {}2λα()Ð1 α Ð1 π = GRk∫ exp Ðik 0 + h /4 (21) a bent silicon crystal with the curvature radius R = 1 m x ∞ λ λ –11 Ð and L0 = 10 m attains the value of d / ~10 . This estimate is better by three orders of magnitude than for 2 × {}ξ () a Johann–Hamós focusing spectrometer [12]. exp ik p/ Lh/Rx Ð 1 dk , It should be noted that the spectral resolution described by (16) is independent of χ and is the same α α hr where the geometric coefficient G depends on 0, h, L0 for both (σ and π) polarizations of the X-ray radiations. α α ≠ and Lh, 0 = 1/L0 – 1/Rx, h = 1/Lh – 1/Rx, and L0, h Rx. The degree of the monochromaticity of the back- The geometrical condition of focusing by a thick scattered wave equals strongly bent crystal has the form 2 λ()α α π ∆λ λ≤ θ∆θ ≅ ()∆θ ≤ χ 1/ 0 + 1/ h /2 +01/4 B = . (22) / cot hr. (17) It is seen from Eq. (22) that the conditions of focus- ∆λ λ Estimate (17) for / can also be obtained under the ing by strongly and weakly bent crystals are different. λ2 ∆λ ≥ assumption that the coherency length is Lcoh = / At the deformation gradient B ≥ 1014 mÐ2 and R ≤ 0.1 m, Λ λ | ||χ | x = / C hr . the source should be very close to the crystal surface. At λ L0 = 1 cm and = 1.54 Å, an X-ray wave dynamically scattered in the crystal bulk is focused at a distance of FOCUSING OF X-RAY WAVE BY AN ELASTIC L cm from the crystal. MOSAIC CRYSTAL h = –1.2499984 With due regard for Eq. (22), we obtain the follow- Consider an elastic mosaic crystal in which the ing equation for the intensity described by Eq. (21): parameter obeys the condition |ν| Ӷ 1 (a strongly bent ()ξ 2π3 χ ()λ2 Ð1 crystal). The bending parameter in backscattering Ih p = G hr 32 B equals +∞ 2 (23) νπ≈ 2χ /16λ2 B (18) × {}ξ () hr ∫ exp ik p/ Lh/Rx Ð 1 dk . and, for the crystals with the deformation gradient B ≥ Ð∞ 14 Ð2 |ν| Ӷ χ –6 ≅ 10 m , the condition 1 is satisfied at hr ~ 10 At the finite integration limits (from –keff to +keff λ ≥ π λ and 1 Å. 2 X/ L0, where X is the size of the region illuminated The amplitude reflection coefficient of a plane-wave by an incident beam), the integral in Eq. (23) is not a harmonic from a thick (semi-infinite) crystal is π/4 π/2 θ () σ ()π 1/2 ()2 {}Φ()1/2 Rk = i h i /8B exp t0 /2 1 Ð Ðt0/2 , (19) –1 where

σ πχ λ θ ≥ πχ()1/2 λθπ()≅ h = C hr/ sin2 hr /2 /2 , –3 ()1/2 ≤ π∆θ λ t0 = Ði/4B k, k 2 / , t –5 Φ()t = 2()π Ð1/2∫ exp()Ðx2 dx 0 –7 is the probability integral. Putting t /21/2 Х π in Eq. (19), we can determine the 0 –9 optimum size of Leff for a strongly bent crystal along the X-axis, ≅≅∆θ ()1/2λ –11 Leff 2L0 2L0 2 B . (20) log d λ/λ The intensity distribution in the focus is determined by the integral over the plane-wave harmonics Fig. 3. Logarithm of the spectral resolution described by Eq. (15) as a function of the angle θ: a < L λ/x , L Ӷ 2 0 eff h I ()ξ = E ()ξ θ |χ | × –6 h p h p Rxsin , Rx = 1 m, and L0 = 2 m. Si (444), hr = 0.94 10 .

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 178 TCHEN

δ-like integral any more. Thus, finally, we arrive at and, it follows from (29) that, at λ = 1.54 Å, X = 10–4, 14 Ð2 2 3 2 Ð1 L = 0.4, and |B| = 4 × 10 m , we have I ()ξ = G π χ ()8λ B 0 h p hr ∼ Ih, weak/Ih, strong 10. ×ξsin()k /()L /R Ð 1 /()ξ /()L /R Ð 1 2, (24) p eff h x p h x Obviously, all the results obtained in this study are also ≠ λ Lh Rx. valid for thermal neutrons with a wavelength of ~ 1 Å. The main intensity maximum in the focus is () π5 χ 2 2 2 λ4 CONCLUSIONS Ih 0 = hr X L0 G /2 B . Thus, we have obtained for the first time an analytical The angular halfwidth of the maximum determining the expression for the size of the region of diffraction reflec- diffraction size of the focus is tion on the crystal surface in backscattering. It follows ∆ξ ≈ λ () p Lh/Rx Ð 1 L0/2X. (25) from this expression that the aperture ratio of the crystal Equation (23) at L = 1, L = 10–3, X = 10–2, R = 0.1 m, optics in Bragg backscattering essentially (by two to 0 h x three orders of magnitude) increases in comparison with λ ∆ξ –8 and = 1.54 Å yields p ~ 10 m. The focus width is the aperture ratio attained in other focusing schemes and dependent on the deformation gradient B and is propor- becomes comparable with the aperture in the Johann tional to |αB – 1|, where α is determined by the focus- scheme. The advantage of the backscattering scheme is ing geometry and the radiation characteristics. The the minimum aberrations of the reflected beam. The height of the intensity peak given by Eq. (24) is analysis of the spectral characteristics of the diffracted inversely proportional to B. beam in backscattering showed that, theoretically, it is Now, consider a crystal of a finite thickness T < Λ. possible to attain a resolution as high as dλ/λ ~ 10–11. The amplitude reflection coefficient is equal to Moreover, we also showed that weakly bent crystals pro- 1/2 1/2 1/2 vide higher intensity at the focus than strongly bent ones. Rk()= Ðii() ()πq/4a ()/2B tanθΦ[ ()Ðt /2 1 According to (29), the intensity ratio for these crystals is Φ()]1/2 ()2 proportional to the deformation gradient B. Ð Ðt0/2 exp Ðt0 /2 , π χ λ θ –1 where q = Ca h( sin ) is the coefficient of reflec- REFERENCES tion from an atomic plane, a is the interplanar spacing, 1. K. Kohra and M. Matsushita, Z. Naturforsch. A 27, 484 θ 1/2 θ ≈ and t1 = t0 + 4T cot (–iB) . In backscattering, cot (1972). ∆θ ≤ |χ |1/2 hr , i.e., is less by two to three orders of mag- 2. O. Brummer, H. R. Hoche, and J. Nieber, Phys. Status nitude than at θ < π/2. Assuming that T < π/4λB, we can Solidi A 53, 565 (1979). expand the difference between the probability integrals 3. A. Caticha and S. Caticha-Ellis, Phys. Rev. B 25, 971 into the Taylor series. Limiting the consideration to the (1982). first derivative of the function Φ(t), we have 4. V. I. Kushnir and É. V. Suvorov, Pis’ma Zh. Éksp. Teor. Fiz. 44, 205 (1986) [JETP Lett. 44, 262 (1986)]. Φ()Φ1/2 ()1/2 Ðt1/2 Ð Ðt0/2 5. T. Tchen, V. A. Bushuev, and R. N. Kuz’min, Zh. Tekh. (26) Fiz. 60 (10), 60 (1990) [Sov. Phys. Tech. Phys. 35, 1148 ≈ ()π Ð1/2 ()2 ()1/2 Ð 2 exp Ðt0 /2 t1 Ð t0 /2 . (1990)]. Then the amplitude reflection coefficient has the form 6. H. H. Johann, Z. Phys. 69 (3-4), 185 (1931). 7. K. T. Gabrielyan, F. N. Chukhovskiœ, and Z. G. Pinsker, () ()2 Rk = iqexp Ð t0 T/a. (27) Zh. Tekh. Fiz. 50 (1), 3 (1980) [Sov. Phys. Tech. Phys. 25, 1 (1980)]. The intensity I (ξ ) in the vicinity of the focus is deter- h p 8. F. N. Chukhovskiœ, Metallofizika 3 (5), 3 (1981). mined from Eq. (21) by substituting there Eq. (27). The 9. D. B. Wittry and S. Sun, J. Appl. Phys. 67, 1633 (1990). intensity ratio for thin (T < Λ) and thick (T ӷ Λ) crystals is 10. W. Z. Chang and D. B. Wittry, J. Appl. Phys. 74, 2999 ∼ χ 2 π (1993). IhT, < Λ/IhT, ӷ Λ 32 B hr T / (28) 11. F. N. Chukhovskii, W. Z. Chang, and E. Forster, J. Appl. χ –6 | | ≥ × 14 Ð2 µ and, at T at hr ~ 10 , B 4 10 m , and T ~ 10 m, Phys. 77, 1843 (1995). equals T ~ 0.4. 12. K. T. Gabrielyan, F. N. Chukhovskiœ, and D. I. Piskunov, Expression (28) allows the estimation of an admis- Zh. Éksp. Teor. Fiz. 96 (3), 834 (1989) [Sov. Phys. JETP χ 69, 474 (1989)]. sible crystal thickness at the given hr and B 13. V. G. Kon, I. V. Kon, and É. A. Manykin, Zh. Éksp. Teor. <<π ()χ 1/2 Λ T /32B hr . Fiz. 116 (3), 940 (1999) [JETP 89, 500 (1999)]. 14. T. Tchen, Pis’ma Zh. Tekh. Fiz. 27 (21), 1 (2001) [Tech. To increase the intensity at the focus, one has to use a Phys. Lett. 27, 889 (2001)]. weakly bent and not a strongly bent crystal. The inten- 15. Z. G. Pinsker, X-ray Crystal Optics (Nauka, Moscow, 1982). sity ratio is estimated as ∼ π 2 Ih, weak/Ih, strong 8 B / keff , (29) Translated by L. Man

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

THEORY OF CRYSTAL STRUCTURES

Quasi-Two-Dimensional Crystalline Clusters on a Sphere: Method of Topological Description A. M. Livshits and Yu. E. Lozovik Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 142092 Russia e-mail: [email protected] Received February 2, 2000; in ﬁnal form, September 4, 2001

Abstract—The formation of a “quasicrystal” on a closed surface has been considered for the Thomson problem on the arrangement with the lowest energy of N Coulomb charges on a sphere. The stable and metastable states of the system of charges with the charge number N = 2Ð100 and the symmetry groups of the corresponding con- ﬁgurations have been determined. The structure and possible structural transitions between the system states are described in terms of the introduced notion of a closed quasi-two-dimensional triangular lattice with topological defects. The graph of lattice defects is deﬁned. A method for classifying the system in terms of the charge and the arrangement of topological defects in the lattice is suggested and extended to the case of an arbitrary lattice. The use of the model is considered on various physical examples, in particular, on a closed hexagonal lattice with disclinations in fullerenes. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION closed hexagonal lattice with topological defects can be used to describe fullerenes and other carbon clusters Cn Recently, we have seen an ever increasing interest in having more complicated structures. Carbon clusters of mesoscopic systems and clusters [1, 2]. A large number various dimensions have already been obtained experi- of real electron and semiconductor mesoscopic systems mentally [10Ð13]. A closed triangular lattice can be is described by the model of repulsing particles in two- formed during crystallization of a cluster of repulsing and three-dimensional confining potentials [3–8]. One particles on a closed surface or in a confining potential. also uses various repulsive (Coulomb, dipole, logarith- In the latter case, a structure of concentric shells can be mic, etc.) potentials. All the known mesoscopic sys- formed [4], with each shell being represented by a tems described by such models show the following closed triangular lattice. Because of the topological common features: (1) formation of a shell structure in properties of a closed surface, any quasi-two-dimen- the equilibrium state and (2) the “orientational melting sional lattice always has some topological defects (dis- of shells” during heating, i.e., the loss of the orienta- clinations), with the total topological charge M of all the tional order of neighboring shells reducing to the rota- defects in each type of the closed lattice being constant. tion of the “frozen” shells with respect to one another In particular, in a closed triangular lattice, Mtr = +12. and then, with a further increase of the temperature, the Since the total topological charge of the defects is fixed, loss of the radial order and the destruction of the shell it is possible to describe the lattice by enumerating all structure. Thus, we arrive at the notion of the “solid” its defects (the so-called lattice index) [14]. Below, we and “liquid” states of a cluster. Of course, this notion fully characterize a closed lattice by a “graph of can be applied to mesoscopic systems only condition- defects” or a d-graph defining the charges of defects ally, e.g., because of the existence of essential “het- and their mutual arrangement. The “distances” between erophase fluctuations” [9] associated with the jump of the defects are introduced in a way used in the theory of the system from the global potential-energy minima graphs. into the local ones. Since a cluster consists of a limited number of particles, it possesses no translation invari- We consider the problem of the structure of closed ance, which, in turn, can result in the appearance of lattices on an example of the Thomson problem fivefold symmetry axes. In this article, we attempt to [15−17] of the optimum arrangement of point charges rigorously describe the structure of a mesoscopic clus- on a sphere, in which the Coulomb energy of the ter in the solid state. N | r | –1 charges, E(N) = ∑ij> ( ri – j ) , is minimized. The Considering the cluster structures, we can single out model describes a number of real physical systems, in a class of clusters that can be represented by quasi-two- particular, a system of ions or bubbles with electrons in dimensional (quasi-2D) closed lattices. A closed lattice a liquid helium cluster, the rarefied system of electrons is a lattice that can be inscribed into a simply connected (or holes) in a spherical “point” in a semiconductor, and closed two-dimensional surface with the unavoidable a system of ions in a three-dimensional trap (3D-trap) formation of some topological defects. The model of a with the potential drastically increasing in the vicinity

180 LIVSHITS, LOZOVIK of the boundary. Indeed, according to the Earnshaw The equilibrium configurations obtained for the theorem, the equilibrium charges in a three-dimen- Thomson problem are listed in Table 1. At N ≤ 90, we sional potential well are located on the surface. Because also indicate the solutions of the Tammes problem (the of the medium polarization, the charges, ions, or bub- data from [23] at N ≤ 12 and the data from [24] at 13 ≤ bles with electrons in a spherical helium cluster are in N ≤ 90). ext the effective confining potential U (r). The shape of The configuration energies are obtained with an this potential is similar to that of a potential well accuracy considerably higher than in [17, 19, 25, 26], with rounded-off edges in the center of a cluster which is of great importance for the selection of the ext | ξ 2 U (r) 0 ~ r and in the vicinity of the surface global minimum from a large number of local ones ext | ξ –1 ξ ≈ U (r) 1 ~ (r – 1) , where 0.03. In the zero approx- (which often possess only slightly different energies) at imation, the potential Uext(r) degenerates into a rectan- charge numbers N ~ 100. This, in turn, provides the gular potential well. A similar effective potential is also determination of some other configuration characteris- formed for a three-dimensional charged semiconductor tics with a considerably higher accuracy. Some of the or semimetal quantum dot. The classical mode of a data obtained agree quite well with the data indicated quantum dot is characteristic of the parameter range, by Edmundson [25], who analyzed all possible config- where the characteristic thermal length of the de Bro- urations for the Thomson problem at charge numbers ≤ glie wave of electrons, λ ~ ប/ mkT , is significantly N 60 and N = 72, 92, 100. The configurations we determined at N = 55, 56, and 92 differ from those less than the average distance between electrons. obtained in [25] and correspond to lower minima. The To construct a unique closed triangular lattice of discrepancies in the Föppl indices for the configura- charges, we use the edges of a convex polygon con- tions at N = 19 and 43 are insignificant and are structed on the chargesÐvertices. Usually, such a poly- explained by the higher accuracy of our calculations. gon has only triangular faces, because, as a rule, the The symmetry data, the Föppl indices, and other char- configurations with nontriangular faces are unstable. acteristics of equilibrium configurations (except for This is confirmed by the following symmetric configu- energy) at charge numbers in the range 60 < N < 100 are rations of the charges: a cube with N = 8, a dodecahe- obtained in our study for the first time. Earlier [16, 17, dron with N = 20, and a “buck ball” with N = 60. 27, 28], energies at charge numbers N ≤ 100 were cal- There is an alternative method of describing the culated with different accuracies. It was also indicated equilibrium structures of point charges on a sphere. The that a convex polyhedron built on the chargesÐvertices configuration of the charges on a sphere can be repre- usually has 12 verticesÐpentamers and (N Ð 12) verti- cesÐhexamers [17, 25, 26]. Altshuller et al. [17] sented by a system of coaxial rings of charges ≤ ≤ [15, 18, 19]. In this case, the structural changes occur- obtained the list of configurations at 13 N 100 ring with an increase of N correspond to the gradual fill- which contradicts the above statement. The results ing of the rings. This approach is justified only for a obtained in the present study considerably differ from small number of charges, N ~ 20, whereas at large N the data indicated by Altshuller (cf. Table 1 in [17] and values, the approach suggested in this article seems to Table 2 in the present article). be preferable. In 1957, Leech showed [29] that the solution of the There is a large class of variational problems [20] variational problem of the search for the minimum of topologically close to the Thomson problem (e.g., the N Ðn U(N, n) = ∑pq≠ r pq at N = 2–6, 12 is independent of Tammes [21] problem on the determination of the con- the exponent n (n = 1, 2, …, ∞) [29]. It should be indi- figuration of N particles on a sphere such that the min- cated that at all the other N values, the solutions of the imum angular distance between the particles would be ∞ ≤ ≤ Thomson (n = 1) and the Tammes (n = ) problems do maximal). It should be indicated that at 12 N 100, not coincide (see Table 1). In particular, the Thomson the Thomson and the Tammes problems usually have problem has many more symmetric solutions. The only different solutions. global energy minimum for the Thomson problem in the range of N under consideration having no symmetry CALCULATION FOR THE THOMSON elements is found at N = 61. PROBLEM The configurations at N = 25, 33, 47, and 79 correspond to the symmetry group C and, thus, have no The energy minima were determined by the gradient 1h rotational symmetry axis. Therefore, in order to write descent method [22], which for our system is written in the Föppl indices for these configurations (Table 1), we the following way [16]: had to use the direction normal to the symmetry plane. r r' = ()r + γ F / r +,γ F Thus, the Föppl index of the configuration at N = 79 is i i i i i i written as follows: 132, 2, 11, 2, 132. In this case, the where Fi is the force acting onto the ith particle. Of all convex polyhedron constructed on the chargesÐvertices the configurations determined, we select the local and has a tetragonal face which is normal to the symmetry the global (possessing the minimum energy) minima. plane. It should also be indicated that the number of

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 QUASI-TWO-DIMENSIONAL CRYSTALLINE CLUSTERS 181

Table 1. Equilibrium conﬁgurations of the system of N charges (N = 2Ð100) on the surface of a sphere N MR G δ Energy L Fo.. ppl G' δ'

20 D∞h 180.00000 0.50000000 Ð 2 D∞h 180.00000 30 D3h 120.00000 1.73205081 Ð 1, 2 D3h 120.00000 40 Td 109.47122 3.67423461 4Tr 1, 3 Td 109.47122 50 D3h 90.00000 6.47469149 2Tr3T 1, 3, 1 D3h, C4v 90.00000 60 Oh 90.00000 9.98528137 6T 1, 4, 1 Oh 90.00000 70 D5h 72.00000 14.45297741 5Tr2P 1, 5, 1 C3v 77.86954 2 80 D4d 71.69415 19.67528786 8P2F 4 D4d 74.85849 3 90 D3h 69.18975 25.75998653 3T6P 3 D3h 70.52878 2 10 0 D4d 64.99562 32.71694946 2T8P 1, 4 , 1 C2v 66.14682 2 11 0.001202 C2v 58.53956 40.59645051 2T8P 1, 2, 4, 2 C5v 63.43495 2 12 0 Yh 63.43495 49.16525306 12P 1, 5 , 1 Ih 63.43495 2 2 13 0.000678 C2v 52.31692 58.85323061 T10P 1, 2 , 4, 2 C4v 57.13670 2 14 0 D6d 52.86609 69.30636330 12P 1, 6 , 1 D2d 55.67057 5 15 0 D3 49.22487 80.67024411 12P 3 C3, C1 53.65785 5 16 0 T 48.93621 92.91165530 12P 1, 3 D4d 52.24440 3 17 0 D5h 50.10802 106.05040483 12P 1, 5 , 1 C2v 51.09033 4 18 0 D4d 47.53440 120.08446745 2T8P 1, 4 , 1 C2 49.55666 2 19 0.000007 C2v 44.90970 135.08946756 F10P 1, 4, 2, 4, 2 , 4 C1h 47.69191 2 2 20 0 D3h 46.09330 150.88156833 12P 1, 3 , 6, 3 , 1 D3h 47.43104 2 2 2 21 0.000067 C2v 44.32038 167.64162240 T10P 1, 2 , 4, 2 , 4, 2 C1 45.61322 2 3 22 0 Td 43.30201 185.28753615 12P 1, 3 , 6, 3 C1 44.74016 7 23 0 D3 41.48111 203.93019066 12P 1, 3 , 1 C1 43.70996 24 0 O 42.06529 223.34707405 6F 46 O 43.69077 10 10 25 0.000041 C1h 39.60981 243.81276030 F10P 1 , 5, 1 C3 41.63446 13 26 0.000074 C2 38.74214 265.13332632 12P 2 C1h 41.03766 5 27 0 D5h 39.94028 287.30261503 12P 1, 5 , 1 C2v 40.67760 9 28 0 T 37.82374 310.49154236 12P 1, 3 C1 39.35514 9 29 0 D3 36.39129 334.63443992 12P 1, 3 , 1 C1 38.71365 14 30 0 D2 36.94228 359.60394590 12P 1, 2 , 1 D3 38.59712 2 2 2 31 0.000103 C3v 36.37311 385.53083806 12P 1, 3 , 6, 3 , 6, 3 C5 37.70983 6 32 0 Ih 37.37736 412.26127465 12P 1, 5 , 1 D3 37.47521 13 13 33 0.000132 C1h 33.69955 440.20405745 F11PH 1 , 7, 1 C3 36.25455 16 34 0 D2 33.27343 468.90485328 12P 1, 2 , 1 C2 35.80778 15 35 0.000012 C2 33.10029 498.56987249 12P 1, 4, 2 C1 35.31846 18 36 0 D2 33.22727 529.12240838 12P 2 D2 35.18973 7 37 0 D5h 32.33243 560.61888773 12P 1, 5 , 1 C1h 34.42241 6 38 0 D6d 33.23648 593.03850357 12P 1, 6 , 1 D6d 34.25066 2 2 39 0 D3h 32.05295 626.38900902 12P 3 , 6, 3, 9, 3, 6, 3 C1 33.48905 2 2 2 40 0 Td 31.91635 660.67527883 12P 1, 3 , 6, 3 , 6, 3, 6, 3 C3 33.15836 2 2 41 0 D3h 31.52783 695.91674434 12P 1, 3 , 6, 3, 9, 3, 6, 3 , 1 C1 32.72909 3 3 42 0 D5h 31.24474 732.07810754 12P 1, 5 , 10, 5 , 1 D5 32.50639 2 2 3 2 43 0.000009 C2v 30.86664 769.19084646 13P 1, 2, 4, 2, 4 , 2 , 4 , 2, 4, 2 C1 32.08447 3 3 44 0 Oh 31.25761 807.17426308 6F 4 , 8, 4, 8, 4 C2 31.98342 15 45 0 D3 30.20718 846.18840106 12P 3 C1 31.32308 15 46 0 T 29.79025 886.16711364 12P 1, 3 C2 30.95916 20 20 47 0.000053 C1h 28.78730 927.05927068 F10P 1 , 7, 1 C1 30.78182 48 0 O 29.68964 968.71345534 6F 412 O 30.76279 16 49 0.000031 C3 28.38659 1011.55718265 12P 1, 3 C2 29.92358 8 50 0 D6d 28.71140 1055.18231473 12P 1, 6 , 1 D6 29.75296

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 182 LIVSHITS, LOZOVIK

Table 1. (Contd.) N MR G δ Energy L Fo.. ppl G' δ' 17 51 0 D3 28.16539 1099.81929032 12P 3 C1 29.36159 17 52 0.000009 C3 27.66987 1145.41896432 12P 1, 3 T 29.19476 2 2 53 0.000005 C2v 27.13694 1191.92229042 3F6P 1, 4, 2, 4 , 2, 4 , C1 28.81390 22, 44, 22, 4 27 54 0.000003 C2 27.02959 1239.36147473 12P 2 C1 28.71692 27 55 0.000007 C2 26.61507 1287.77272078 12P 1, 2 C1 28.26279 38 56 0 D2 26.68290 1337.09494528 12P 2 D2 28.14805 19 57 0 D3 26.70241 1387.38322925 12P 3 C3 27.82668 11 2 11 58 0 D2 26.15523 1438.61825064 12P 1, 2 , 4, 2 , 4, 2 , 1 C1 27.55485 29 59 0.000003 C2 26.17024 1490.77333528 14P2H 1, 2 C1 27.39498 20 60 0 D3 25.95762 1543.83040098 12P 3 C2 27.19283 61 61 0.000018 C1 25.39167 1597.94183020 12P 1 C1 26.87233 12 62 0 D5 25.87987 1652.90940990 12P 1, 5 , 1 C1 26.68343 21 63 0 D3 25.25672 1708.87968150 12P 3 D3 26.48692 32 64 0 D2 24.92001 1765.80257793 12P 2 C1 26.23504 32 65 0.000006 C2 24.52673 1823.66796026 12P 1, 2 C2 26.06983 33 66 0.000012 C2 24.76463 1882.44152531 12P 2 D3 25.94744 13 67 0 D5 24.72726 1942.12270041 12P 1, 5 , 1 C2 25.68398 34 68 0 D2 24.43292 2002.87470175 12P 2 C2 25.46062 23 69 0 D3 24.13651 2064.53348323 12P 3 C1 25.33223 2 8 70 0 D2d 24.29073 2127.10090155 4F4P 1, 2, 4, 2, 4 , 2, 4 , C1 25.17092 2, 42, 2, 4, 2, 1 28 5 71 0.000018 C2 23.80257 2190.64990643 14P2H 1, 2 , 4, 2 C1 24.98791 14 72 0 I 24.49170 2255.00119097 12P 1, 5 , 1 D3 24.92649 36 73 0.000022 C2 22.81041 2320.63388375 12P 1, 2 C1 24.55376 37 74 0.000009 C2 22.96584 2387.07298184 12P 2 C1 24.42088 25 75 0 D3 22.73643 2454.36968904 12P 3 C1 24.30172 38 76 0.000012 C2 22.88571 2522.67487184 12P 2 C1 24.11969 15 77 0 D5 23.28614 2591.85015235 12P 1, 5 , 1 C1 23.99585 26 78 0 Th 23.42634 2662.04647457 12P 3 D3 23.93102 32 23 79 0.000009 C1h 22.63614 2733.24835748 F11PH 1 , 2, 11, 2, 1 C1 23.62399 3 2 2 2 3 80 0 D4d 22.77835 2805.35587598 2F8P 4 , 8, 4 , 8, 4 , 8, 4 , 8, 4 D2 23.54352 40 81 0.000002 C2 21.89175 2878.52282966 12P 1, 2 C1 23.34764 40 82 0 D2 22.20594 2952.56967529 12P 1, 2 , 1 C1 23.19261 41 83 0.000004 C2 21.64623 3027.52848892 14P2H 1, 2 C1 23.08300 41 84 0.000005 C2 21.51267 3103.46512443 12P 1, 2 , 1 D3 23.05173 42 85 0.000005 C2 21.49758 3180.36144294 12P 1, 2 C1 22.77869 43 86 0.000016 C2 21.52160 3258.21160571 12P 2 C1 22.67437 17 24 87 0.000009 C2 21.45649 3337.00075001 12P 1, 2 , 4, 2 D3 22.54666 44 88 0 D2 21.48559 3416.72019676 12P 2 T 22.46788 44 89 0.000001 C2 21.18220 3497.43901862 12P 1, 2 C1 22.31660 30 90 0 D3 21.23028 3579.09122272 12P 3 D3 22.15402 45 91 0 C2 21.10466 3661.71369932 12P 1, 2 46 92 0 D2 21.02582 3745.29163624 12P 2 46 93 0.000002 C2 20.75075 3829.84433842 12P 1, 2 46 94 0 D2 20.95187 3915.30926962 12P 1, 2 , 1 47 95 0.000001 C2 20.71121 4001.77167557 12P 1, 2 12 34 96 0 C2 20.68657 4089.15401006 12P 2 , 4, 2 48 97 0.000001 C2 20.44961 4177.53359962 12P 1, 2 49 98 0.000001 C2 20.42160 4266.72246416 12P 2 33 14 99 0.000002 C2 20.28450 4357.13916313 12P 1, 2 , 4, 2 100 0 T 20.29660 4448.35063433 12P 1, 333 Note: MR is the dipole moment of the system, G is the symmetry group (in Schönﬂies notation), δ is the minimum angular distance between the charges (in degrees), L is the index of the closed lattice, Föppl indicates the Föppl index, Energy indicates the total Cou- lomb energy of the system, G' is the symmetry group used in the solution of the Tammes problem, and δ' is the minimum angular distance between the particles in the Tammes problem.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 QUASI-TWO-DIMENSIONAL CRYSTALLINE CLUSTERS 183 charges located in the symmetry plane in all cases with Table 2. Exceptions from the rule deﬁning the number of symmetry C1h is described by a simple number—5, 7, verticesÐpentamers and verticesÐhexamers in a convex poly- 7, or 11. hedron constructed on the chargesÐvertices at 12 ≤ N ≤ 100

It was indicated earlier that, usually, the configura- NN4 N5 N6 N7 Q tions with tetragonal faces are unstable. In particular, it is well known that a cube does not correspond to the 131102 minimum energy in the Thomson problem. We deter- 18 2 8 8 mined three configurations having cubic symmetry 19 14 5 1 (at N = 24, 44, 48) and possessing tetragonal (in this case, square) faces. The same configurations are also 2111010 indicated in [25] but, surprisingly, are not indicated in 24 24 6 the later study [17]. The configuration at N = 70 is 25 14 11 1 described by the symmetry group D2d and has four tet- 33 15 17 1 1 ragonal faces. Two of three fourfold rotation reflection 44 24 20 6 axes pass through the centers of the tetragonal faces, whereas the third rotation reflection axis transforms 47 14 33 1 these four faces into one another. 48 24 24 6 Configurations possessing higher symmetries can 53 18 35 3 have higher energies than the low-symmetric ones. 59 14 43 2 Thus, at N = 42, the local minimum (E = 70 20 50 4 732.15182672) has the symmetry I , whereas the global h 71 14 55 2 minimum has a lower symmetry D5h. Table 3 lists the local minima of the system of point charges on a sphere 79 15 63 1 1 at N = 92. Of nineteen configurations listed in Table 3, 80 16 64 2 the minimum possessing the highest symmetry Ih is 83 14 67 2 also characterized by the highest energy. Note: N is the number of charges; N4, N5, N6, and N7 are the numbers of vertices incident to 4, 5, 6, and 7 edges, respectively; and Q is the number of tetragonal faces. DISCUSSION OF RESULTS One of the most well-known approaches to the description of the charge distribution on a sphere is that natural method used to describe the configurations is from the classical study performed by Thomson [15], the consideration of the characteristics of this polyhe- who considered that the structures consisted of a small dron. Below, we study such characteristics for a poly- number of charges. According to Thomson, the charges hedron uniquely determining the structures of all the form coaxial rings on a sphere, whereas the structural configurations. changes occurring with an increase in the charge num- In fact, a convex polyhedron constructed on ber N in the system correspond to the gradual addition chargesÐvertices is the implementation of the idea of a of charges to different rings (Fig. 1). A consistent for- closed two-dimensional triangular lattice with topolog- mal description within the framework of the “ring ical defects. The concept of a closed lattice with topo- approach” was developed by Föppl [30]. The axis of the logical defects can be used to describe structures of var- coaxial rings is taken to be that of the highest symme- ious systems, including fullerenes and a large number try, whereas the particles are considered to be located of clusters consisting of repulsing particles in confining on one ring only if they all are located in the same plane potentials. It is well known that, for the repulsive poten- normal to this axis. This approach is quite justified with –n a small number of charges because the rings of charges tials U ~ r (n = 1, 3, 4, …), a triangular lattice is stable are well distinguishable. With an increase in the charge in the plane and possesses the minimum energy as com- number N, increasingly less information on the real pared with the energies of the lattices of all the other charge configuration can be extracted from the Föppl types [31, 32]—square, hexagonal, etc. Thus, one can expect that the triangular lattice would also be stable in index. Thus, various local minima are often described 〈 〉 Ӷ the region rij R on a curved surface, where R is the by the same indices (see Table 3). The notion of rings 〈 〉 itself no longer has any sense because the distances curvature radius of the surface and rij is the average between the rings can be several times less than the dis- distance between the particles. However, if the lattice is tances between the charges in the individual rings. Nev- considered on the whole sphere surface, i.e., if the sur- ertheless, the Föppl indices can successfully be used for face is closed, it possesses a number of specific features solving a number of problems. based on the topological differences between a sphere (a closed simply connected surface) and a plane. Since the charges in the Thomson problem are located on a sphere, one can always construct a convex In order to construct the lattice on a sphere, one can polyhedron on chargesÐvertices if N ≥ 4. Thus, another use several methods, in particular, the slightly modified

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 184 LIVSHITS, LOZOVIK

Table 3. The Thomson problem: local minima at number of closest equivalent particles. If a plane lattice possesses charges N = 92 a symmetry axis of the nth order, then, in moving .. around the disclination core having the charge m, the G Fo ppl Energy phase φ is changed by a value of δφ = 2πm/n. Thus, a 46 D2 2 3745.29163624 particle surrounded by five and not six closest neigh- 46 bors (a pentamer) in the plane triangular lattice is a core C2 2 3745.32218333 46 of a disclination with the charge m = +1, a particle with C2 2 3745.32555284 four nearest neighbors, a core of disclination with the 46 C2 2 3745.33860835 charge m = +2, etc. (Figs. 2a, 2b). 92 C1 1 3745.36351633 Consider the equilibrium and the quasi-equilibrium 18 D5 1, 5 , 1 3745.36772048 configurations of charges on the sphere surface (at all 92 ≥ C1 1 3745.38100894 N 4) as quasi-two-dimensional closed triangular lat- 92 tices with topological defects. A closed lattice is under- C1 1 3745.38177220 46 stood as a lattice inscribed into any closed simply con- C2 2 3745.38212625 nected surface (not necessarily convex). Hereafter, we 92 C1 1 3745.38297241 use the following notation for defects: a pentamer is a 92 C1 1 3745.38930839 P-disclination (m = +1), a tetramer is a T-disclination 92 (m = +2), a trimer is a Tr-disclination (m = +3), and a C1 1 3745.38943469 10 11 20 heptamer is an H-disclination (m = Ð1). In terms of C2 1, 2 , 4, 2 , 4, 2 , 1 3745.39152813 topology, a tetragonal face is also a disclination (m = +2), 92 C1 1 3745.39598792 which differs from a conventional disclination only by 46 C2 2 3745.41663888 the absence of a particle in the disclination core (the 31 7 7 31 face center) (Fig. 3a). Hereafter, we call such a defect C1h 1 , 2, 1 , 12, 1 , 2, 1 3745.48063056 92 of the closed triangular lattice a “focus” or an F-discli- C1 1 3745.48989730 nation. Formally, defects of this type can be excluded 92 C1 1 3745.50315699 from the lattice. With this aim, one has to add an arbi- 3 2 2 2 3 Ih 1, 5 , 10, 5 , 10 , 5 , 10, 5 , 1 3745.61873913 trarily chosen edge to each tetragonal face. As a result, the modified lattice acquires two P-disclinations in the Note: For notation see Table 1. neighboring vertices instead of each of the F-disclinations, whereas the total topological charge remains unchanged (Fig. 3b). Nevertheless, we preserve here Voronoi method for constructing lattices on curved 2D- the notation of an F-disclination because it is very con- surfaces. As will be shown later, the method used for venient (see below) for describing the structure of a the construction does not affect the basic properties of closed lattice, in particular, the lattice symmetry. Figure the closed lattice. Here, we construct the lattice using 4 shows some charge configurations characterized by the edges of a convex polyhedron with the vertices tetragonal faces. occupied by charges.

LATTICE DEFECTS BASIC PROPERTY OF A CLOSED LATTICE Consider defects of a plane lattice (for details, see, In a macroscopic plane lattice, the formation of dis- e.g., [33]) which can have two types of defects—dislo- clination is low probably because of the high energy of cations and disclinations. Disclinations break the sym- arising elastic deformations. A quasi-two-dimensional metry of the directions of the vectors connecting the closed lattice always contains disclinations with the

345678

9 10 11 12 13 14

Fig. 1. Ring structures at N = 3Ð14. The distorted ring structure at N = 13: {1, (6), (5), 1}.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 QUASI-TWO-DIMENSIONAL CRYSTALLINE CLUSTERS 185

(a) (b) (a) (b) (c)

F F P

K = 6 P K = 5 Fig. 3. F-disclination in a triangular lattice: (a) formation of an F-disclination (m = +2); (b) decomposition of an F dis- Fig. 2. Disclinations in a plane triangular lattice δφ = clination into two P-disclinations; and (c) the virtual addi- 2πm/6: (a) the disclination with m = +1 and a wedge δφ = tion of a particle to the core of each F disclination in the π +2 /6 cut out from the lattice; (b) the disclination with m = construction of the d graph. The distance LPF between two Ð1 and a wedge δφ = –2π/6 inserted into the lattice. disclinations equals three. total topological charge of M defects defined as is seen from Table 1, most structure defects are of the 12P M = ()N Ð N +++2()N Ð N 3()…N Ð N type. Then, we can also introduce the characteristics +1 Ð1 +2 Ð2 +3 Ð3 reflecting the arrangement of defects in a closed lattice. being constant and independent of the number of particles and of the method of lattice construction (where Ni is the number of disclinations in the lattice with the STRUCTURE OF A CLOSED LATTICE. GRAPH topological charge m = i). In particular, in a close trian- OF LATTICE DEFECTS gular lattice The topology of a closed lattice is reflected by its Mtr = 12, graph G or a graph of edges of a convex polyhedron constructed on the charges located on a sphere. The in a closed hexagonal lattice, Mhex = 12, and in a closed square lattice M = 8. Consider the proof for a closed symmetry of the charge configuration described by the q point symmetry group is either lower or the same as the triangular lattice. The numbers of faces, vertices, and symmetry of the graph of a closed lattice constructed edges of a convex polyhedron, F, V, and E, respectively, for the given configuration. This relationship is fulfilled are related by the Euler relationship V + F Ð E = 2. for a closed lattice of an arbitrary structure including Three edges are incident to each face of a closed trian- the lattice containing F-disclinations. If one were to gular lattice with each edge being tangential to two construct a closed lattice by excluding F-disclinations faces. Performing summation over the faces, we have and adding arbitrarily an edge to each tetragonal face of 3F = 2E. Every vertex of a closed triangular lattice is the polyhedron, one would arrive at a situation where incident to six edges. If a vertex is a disclination with the symmetry of the configuration would be higher than the charge m = +1, this vertex is incident to five edges the symmetry of the corresponding graph, which would (i.e., one edge less). If the disclination charge is m = Ð1, be very inconvenient for the classification. the vertex is incident to seven edges (one edge more), etc. Upon summing up all the vertices of the triangular Now, determine the distance d(u, v) between two lattice with disclinations, with due regard for the fact that vertices, u and v, of the closed lattice as a length of the each edge was summed up twice, we obtain 6V – Mtr = 2E. shortest simple chain connecting these vertices. Now, construct the graph of defects (the d-graph) in the fol- Now, expressing Mtr from the above formulas, we arrive at the sought equation. lowing way. Take the vertices of the closed lattice (disclinations) as the vertices of the d-graph. Connect these verticesÐdisclinations by edges and bring each edge into LATTICE INDEX correspondence with the distance between the corre- The lattice index is introduced to describe the types sponding defects. Constructing the d-graph for a structure of defects in the lattice and their numbers. The exam- containing F-disclinations, we can virtually add a particle ples of such indices are 12P, 2T8P, and 4F. The number into the core of each F-disclination (Fig. 3c). before the defect symbol indicates the number of Since the distance between any two vertices is deter- defects of the given type in the lattice. Thus, index mined uniquely, it is obvious that, for the given config- 12P signifies that the lattice contains 12 P-disclina- uration of the charges forming the closed lattice, the d- tions, while index 2T8P indicates that the lattice has graph can also be constructed uniquely. The opposite 2 T- and 8 P-disclinations, and so on. A PH-dipole is question of whether the graph of defects determines the considered as one defect—a dislocation (m = 0). The structure of the closed lattice uniquely or nonuniquely FH-complex formed in the systems with N = 33 and 79 is less trivial. We compared the d-graphs of all the local can also be considered as one complex defect (m = +1). As minima determined and established that, in the range of

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 186 LIVSHITS, LOZOVIK

19 24 44

48 53 70

79 80 83 Fig. 4. Structures containing F-disclinations at N = 19, 24, 44, 48, 53, 70, 79, 80, 83. At N = 83, the local minimum E = 3027.558629 (the point group C1h) is seen. The remaining figures show the configurations corresponding to the global minima of the systems. N values 4 ≤ N ≤ 100, the d-graph uniquely determines The matrix of defects has an infinite number of invari- the structure of the closed lattice. ants of the form To a graph of polyhedron edges there corresponds Inm(), = V ()n V ()m , the symmetric matrix A of dimensions N × N; then, ∑ i j ij, aij = 1 if the vertices i and j are connected by an edge, and a = 0 if they are not connected. The graph of n ij where Vi(n) = ∑ ≠ dij . The numbers n and m are arbi- defects can be written with the aid of the symmetric ji d-matrix (k × k), where k is the number of defects in the trary real numbers. One can also introduce a number of structure. To the nondiagonal matrix elements there quantitative characteristics which are the functions of correspond the pairs of distances between the disclina- the d-matrix, e.g., the average distance between the tions. Now, write the symbols of the defects on the defects in the lattice 2 diagonal of the d-matrix. Since all the charges are M = ------∑d equivalent, the operation of defect permutation in the kk()Ð 1 ij d-matrix does not change the structure of the d-graph ij> and the closed lattice. The permutation of the K and and the dispersion of these distances L defects is defined as 2 2 2 D = ------()∑dij Ð M ,  ' kkÐ 1 dKK = dLL ij>  ' where k is the number of defects. In the systems con- dLL = dKK taining different types of defects, one can also intro-  d' ==d' d duce the average distance and dispersion for defects of [] [] KL LK KL each type separately, e.g., M and M . dij ' = swapdij :  (1) PP FF KL,  ()≠ , diL' ==dLi' diK iLK Figure 5 shows the matrices of defects for two dif-  ferent local energy minima in the Thomson problem at  ' ' ()≠ , diK ==dKi diL iLK N = 92. The local minima (the second and third rows in  Table 3, respectively) are described by the same Föppl ' (), ≠ , dij = dij ij LK. indices and the same symmetry groups. At the same

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 QUASI-TWO-DIMENSIONAL CRYSTALLINE CLUSTERS 187

(a) charges on the surface of a sphere. It is shown that the triangular lattice on a closed simply connected surface includes topological defects—disclinations—of different charges, with the total topological charge of defects, M = 12, being constant and independent of the charge and location of individual defects. The equilibrium and nonequilibrium configurations are considered at charge numbers in the range N = 2Ð100. The methods for classification and compact description of the structures are suggested. A graph of defects or a d-graph of the lattice is defined and used to describe the mutual location and charge of defects in the lattice. Thus, the description of the crystal structure of N charges on a closed surface is (b) reduced to determining the d-graph or the d-matrix of the lattice, k × k, where the number of defects in the lattice k is limited. In particular, in the Thomson problem, for most of the configurations we obtain k = 12. The model of Coulomb charges on the surface of a sphere describes a number of physical systems, including the 3D-system of ions in a trap with an abrupt confining potential, a system of electrons in a semiconductor 3D-quantum dot, a multicharge bubble with an electron in liquid helium (in the classical limit), and a system of ions in a helium cluster retained by image forces. At present, a large number of various objects are known and experimentally observed that are character- Fig. 5. The d-matrices of two “similar” minima in the ized by one common property: From the topological Thomson problem: (a) the d-matrix of the local minimum at 46 standpoint, these systems can be considered as sets of N = 92 (the symmetry C2, the Föppl index 2 ); (b) the points distributed over the sphere surface. Such objects d-matrix of another local minimum at N = 92 (the symme- 46 are clusters consisting of particles interacting according try C2, the Föppl index 2 ). to different (Coulomb, dipole laws) laws, atomic clusters, multiatomic molecules, fullerenes, spherical time, their d-matrices are different and cannot be viruses, etc. It is suggested that such structures be con- reduced to one another. sidered within the unified approach as quasi-two- Since the interaction potential between the charges dimensional closed lattices with topological defects of is not used in the construction of the graph of a closed various charges. The structure of the lattice and the lattice, the formalism of the d-graph and the d-matrix character of the defects in it are determined by various can be used not only for writing and comparing differ- mechanisms, e.g., by valence (in fullerenes and other ent local and global minima (the solutions of the Thom- atomic and molecular clusters), mutual charge repul- son and other close problems) but also for comparing sion resulting in the formation of a triangular lattice, the solutions of various problems. etc. For each type of closed lattice, the total charge of all the disclinations is an invariant. Thus, in the closed HEXAGONAL LATTICE triangular lattice and in the closed hexagonal lattice (fullerenes), the total charge of all the defects is M = 12. A model of a closed hexagonal lattice can be used to However, in the former case, disclinations are verticesÐ describe fullerene structures. A graph of a closed hex- nonhexamers, whereas in fullerenes, disclinations are agonal lattice is dual with respect to the graph of a nonhexagonal faces. A number of systems, e.g., ions in closed triangular lattice in which the centers of discli- traps and multishell carbon clusters (“onions”), can be nations in the closed hexagonal lattice are faces and not considered to be systems of concentric closed lattices. vertices. Thus, to a closed hexagonal lattice of fullerene The development of nanotechnology allows one to C60 there corresponds the closed triangular lattice of the expect the creation of new substances with molecular local minimum (N = 32, Ih) in the system of Coulomb structures in the form closed lattices. charges on a sphere. REFERENCES CONCLUSION 1. Proceedings of VIII International Symposium on Small The notion of a quasi-two-dimensional closed trian- Particles and Inorganic Clusters, ISSSPIG 8, Copen- gular lattice is introduced to describe the structure of hagen, Denmark, July 6, 1996.

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2. Proceedings of the Conference “Mesoscopic and 18. H. A. Munera, Nature 320, 597 (1986). Strongly Correlated Electron Systems,” Chernogolovka, 19. L. T. Wille, Nature 324, 46 (1986). 1997, Usp. Fiz. Nauk 168 (2), 135 (1998) [Phys. Usp. 41, 127 (1998)]. 20. T. W. Melnyk, O. Knop, and W. R. Smith, Can. J. Chem. 55, 1745 (1977). 3. Yu. E. Lozovik, Usp. Fiz. Nauk 153, 356 (1987) [Sov. Phys. Usp. 30, 912 (1987)]. 21. P. M. L. Tammes, Recl. Trav. Bot. Neerl. 27, 1 (1930). 4. Yu. E. Lozovik and V. A. Mandelshtam, Phys. Lett. A 22. A. A. Samarskiœ and A. A. Gulin, Numerical Methods 165, 469 (1992). (Nauka, Moscow, 1989). 5. V. M. Bedanov and F. M. Peeters, Phys. Rev. B 49, 2667 23. A. L. Mackay, J. L. Finney, and K. Gotoh, Acta Crystal- (1994). logr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystal- 6. V. Shweigert and F. M. Peeters, Phys. Rev. B 51, 7700 logr. A33, 98 (1977). (1995). 24. D. A. Kottwitz, Acta Crystallogr., Sect. A: Found. Crys- 7. Yu. E. Lozovik and E. A. Rakoch, Phys. Rev. B 57, 1214 tallogr. A47, 158 (1991). (1998). 25. J. R. Edmundson, Acta Crystallogr., Sect. A: Found. 8. A. I. Belousov and Yu. E. Lozovik, Pis’ma Zh. Éksp. Crystallogr. A48, 60 (1992). Teor. Fiz. 68, 817 (1998) [JETP Lett. 68, 858 (1998)]. 26. J. R. Edmundson, Acta Crystallogr., Sect. A: Found. 9. A. I. Belousov and Yu. E. Lozovik, Eur. Phys. J. D 8, 251 Crystallogr. A49, 648 (1993). (2000). 27. T. Erber and G. M. Hockney, Phys. Rev. Lett. 74, 1482 10. H. W. Kroto, J. R. Heath, S. C. O’Brien, et al., Nature (1995). 318, 162 (1985). 28. E. L. Altshuler, T. J. Williams, E. R. Ratner, et al., Phys. 11. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 163 Rev. Lett. 74, 1483 (1995). (4), 33 (1993) [Phys. Usp. 36, 202 (1993)]. 29. J. Leech, Math. Gazette 41, 81 (1957). 12. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 165, 30. L. Föppl, J. Reine Angew. Math. 141, 251 (1912). 977 (1995) [Phys. Usp. 38, 935 (1995)]. 31. L. Bonsall and A. A. Maradudin, Phys. Rev. B 15, 1959 13. Yu. E. Lozovik and A. M. Popov, Usp. Fiz. Nauk 167, (1977). 751 (1997) [Phys. Usp. 40, 717 (1997)]. 14. A. M. Livshits and Yu. E. Lozovik, Chem. Phys. Lett. 32. G. Meissner and A. Flamming, Phys. Lett. A 57A, 277 314, 577 (1999). (1976). 15. J. J. Thomson, Philos. Mag. 7, 237 (1904). 33. V. A. Likhachev and R. Yu. Khaœrov, An Introduction to the Theory of Disclination (Leningr. Gos. Univ., Lenin- 16. T. Erber and G. M. Hockney, J. Phys. A 24, L1369 grad, 1975). (1991). 17. E. L. Altshuler, T. J. Williams, E. R. Ratner, et al., Phys. Rev. Lett. 72, 2671 (1994). Translated by L. Man

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Crystallography Reports, Vol. 47, No. 2, 2002, pp. 189Ð195. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 224Ð231. Original Russian Text Copyright © 2002 by Yudin, Pisarenko, Lyubchenko, Chudnova, Karygina.

THEORY OF CRYSTAL STRUCTURES

Penrose Mosaic as a Quasistochastic Tree-Graph Lattice V. V. Yudin, T. A. Pisarenko, E. A. Lyubchenko, O. A. Chudnova, and Yu. A. Karygina Far East State University, Vladivostok, 690600 Russia e-mail: yudin@iﬁt.phys.dvgu.ru Received June 30, 2000

Abstract—The Penrose mosaic as the minimum representative of quasicrystals is discussed in terms of generalized planar lattice models. The role of these models is played by Cayley’s tree graphs which, in the general case, are characterized by quasi-random branching. A three-level golden alphabet is deﬁned, and a Penrose mosaic is synthesized with the aid of its highest level. The algebras of the suggested grammar are formulated in an explicit form. It is shown that the statistics of a Penrose mosaic at the level of golden rhombuses belongs to the class of ZipfÐMandelbrot distributions. The algorithm for mapping a Penrose mosaic into Cayley’s tree graphs based on the [2q × 2p] alphabet is also formulated. The problem of the entropy percolation for quasistochastic Cayley’s trees of Penrose mosaics is solved. The entropy percolation of these trees is characterized by an obvious minimum periodicity and, on average, by the invariance principle of the golden entropy. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION our consideration is the transition to the “quartet alpha- Almost fifteen years ago, the Shechtman group stud- bet” at the level of golden (g) rhombuses with due regard for the types of possible contacts—the vertex ied the rapidly cooled Al Mn alloy and obtained unique 86 14 and the edge neighborhoods of g-rhombuses. objects which did not belong to metallic glasses [1]. The diffraction patterns of these objects had ten spots The creation of the grammar is rather difficult [10], related by a fivefold rotation axis forbidden in crystal- but we can simplify it and avoid the identification of the lography. Later, these objects were called quasicrystals. grammar type by considering a Penrose mosaic in a In an attempt to interpret the “pentasymmetry” of these new tree-graph representation based on the quartet diffraction patterns, the Penrose model was used [2, 3]. alphabet, which is equivalent to the algorithm for con- Somewhat later, a family of mosaics corresponding to structing a Penrose mosaic itself—a typical grammati- forbidden symmetries of higher orders was also studied cal function [10]. [4Ð6]. Percolation on Cayley’s trees for a Penrose mosaic The symmetries characteristic of quasicrystals were can be interpreted as a Markovian shift in the tree hier- also observed in studies of nonlinear dynamic systems archies, which is reflected in the probabilistic and [7], and it was shown that the destruction of the separa- entropy measures. This type of percolation is related to trix structure of the phase space caused by the the semigroup class. Generally speaking, percolation ε -perturbation results in the formation of a stochastic can occur along two directions: center periphery. web with a symmetry characteristic of quasicrystals. As The existence of two opposite semigroup percolation follows from [7], a stochastic web is generated by flows can result in a nonzero defect in percolation on recurrent two-dimensional mapping with twisting. By Cayley’s trees. In this case, we arrive at irreversible tree the end of the 1990s, a specific niche including a wide graphs. It is possible to introduce the criterion of quasi- class of quasicrystal objects had been formed. Thus, it crystallinity on Penrose mosaics in terms of entropy was shown that the processes of structure relaxation functionals. Thus, we arrive at the theoretical-probabi- and decomposition of the amorphous state in metallic listic and entropy manifestations of the “golden charac- glasses resulted in the precipitation of star-shaped teristics” of the Penrose mosaic. grains [8, 9]. Consider a Penrose mosaic as the minimum and uni- STATISTICS OF A PENROSE MOSAIC versal representative of the class of quasicrystals. AT THE LEVEL OF A QUARTET Below, we use the linguistic approach and interpret a RHOMBUS-BASED ALPHABET Penrose mosaic as a text written in a certain language. However, we reduce the linguistic approach to the level Various Penrose mosaics are formed by golden of statistical linguistics, where we use the frequency of rhombuses (g-rhombuses) of only two types, consist- the occurrence of various symbols forming an alphabet ing, in turn, of two golden (g) triangles (Figs. 1a, 1c) and, thus, also rank statistics. A characteristic feature of sharing the golden sides. The existence of the level of

1063-7745/02/4702-0189 $22.00 © 2002 MAIK “Nauka/Interperiodica” 190 YUDIN et al. The number of g-rhombuses in the sentences of (a) (b) (e) level III is 5 + 5 (“acute” + “obtuse”) rhombuses, with both components being equally probable. A pair of ten-vertex polygons is constructed from g-rhombuses by applying the join operation (without intersection) along a nonspecific (unit) side. One of the compound ten-vertex polygons possesses (c) (d) (f) a fivefold rotation axis, whereas the other is characterized by dorsal symmetry with a “pentacoordination” point. The role of a nucleus is played by an obtuse g-rhomb. A three-level binary golden alphabet, level III, and their fragments are illustrated by Fig. 1. Using level III, Fig. 1. Three-level alphabet of a Penrose mosaic. we synthesized a Penrose mosaic shown in Fig. 2 with a rather simple grammar III. The algebra of the grammar for alphabet III is as follows: —a fragment shown in Fig. 1e is a star-shaped ten- vertex polygon that has no contacts with other similar polygons; —a joint with a simple intersection of the motifs in Figs. 1e and 1f is allowed with the intersection being an acute g-rhombus; —a pair joint of dorsal ten-vertex polygons occurs with the intersection in the form of two acute and one obtuse g-rhombuses; —the maximum coordination number of the motif surrounded by dorsal ten-vertex polygons shown in Fig. 1e is five; —five joint dorsal fragments can form a pentasymmetry point (but not a fragment shown in Fig. 1e) with five complex intersections. Now, complement the mosaic representation with the probabilistic consideration based on alphabet II. Along with the objects (a pair of g-rhombuses), we also take into account the contact type (adjacency neighbor- Fig. 2. Penrose mosaic. One can see elements of alpha- hood), i.e., the coordination. The Penrose mosaic has bet III, the types of intersections, and several coordina- both the vertex and the edge contacts of g-rhombuses. tion spheres. Below, we write matrix (1) (Fig. 2) of the probability to encounter the letters “rhombuses-type of the contact” averaged over the whole Penrose mosaic. In fact, we g-triangles and, later, of g-rhombuses (Figs. 1b, 1d), deal here with the quartet alphabet “rhombus (q)— indicates the hierarchical nature of the alphabet of the coordination type (p),” [2q × 2p], with the matrix Penrose mosaic. Now, formulate a hypothesis which states that the synthesis of a mosaic is simplified if an v c alphabet of a higher rank based on blocks (sentences) of α ()α v ∨ ()α , g-rhombuses is used. Possibly, the grammar of the hier- 1 0.4285 0.1905 P 1/ c = P0 1 = 0.619 α ()α ∨ ()α (1) archical alphabet and Penrose tiling would be less com- 2 0.261 0.120 P 2/v c = P0 2 = 0.381 plicated and artificial that at the level of g-rhombuses α α [3, 11]. where 1 is an obtuse g-rhombus, 2 is an acute g-rhombus, c is an edge contact, and v is a vertex contact. It is To construct a dual g-basis of Penrose tiling of level III, seen that the sums of the rows of matrix (1) coincide we have to apply the following algebraic rules of lin- with the golden ratio and its complement. The estimate guistics II: α ∨ α ∨ of the Vajda entropy yields HV [P( 1/v c); P( 2/v A “block” or a sentence of level III is chosen as a c)] = 0.472 = Hgold. The corresponding structurization η pair of convex ten-vertex polygons composed in the factor equals 2 = 5.6%, and therefore the statistics of dual basis of g-rhombuses. g-rhombuses is stochastic for about 94.4%. In this

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 PENROSE MOSAIC AS A QUASISTOCHASTIC TREE-GRAPH LATTICE 191 sense, the Penrose mosaic is an almost noise stochastic parquet at the level of alphabet II. The asymptotic probabilistic vectors of the M-chain can be conveniently mapped into a two-level binary α =0.62 α =0.38 probabilistic tree (Fig. 3). The first level of the tree indi- 1 2 cates a division into two g-rhombuses with golden 1.62 probabilities, which results in Hgold(2) = 0.472. At the second level, two more types of contact—the vertex αv =0.48 αc =014 αv αc and the edge ones—are added, which results in the for- 1 1 2 =0.26 2 =0.12 mation of the quartet [2q × p] alphabet. Considering 0.29 0.46 Fig. 3, we can see that the golden ratio of the first-level probability is preserved at the second level in the form 0.63 of the double relationship Fig. 3. Two-level binary tree of asymptotic two- and four- v dimensional probabilistic vectors of a Penrose mosaic. α α /α c -----1 ==------1 1- 0.618∨ 1.618. α α v α c ln 2 2 / 2 r 0 0.4 0.8 1.2 1.6 Then, the tree in Fig. 3 should be recognized as a golden tree on the whole. The estimate of the Vajda –0.8 entropy, H(4) = 0.6688 = Hgold(4), should also be recognized as a golden one in the quartet alphabet. The struc- × η –1.2 tural index in the [2q 2p] alphabet, 4 = 10.66…%, indicates a higher order level of the Penrose mosaic in the quartet alphabet. Obviously, the stochastic compo- –1.6 nent of the Penrose-mosaic organization in this alphabet is about 90%. Compared to the binary alphabet, 2 1 the quartet alphabet [2q × 2p], remaining golden as –2.0 earlier, provides a higher organization of the Penrose mosaic. As is seen from Eq. (1), it is possible to order all four –2.4 conditional probabilities by the degree of their decrease on the matrix [2q × 2p]. In fact, this signifies the transition to the consideration of rank statistics. Consider the –2.8 problem of identifying rank statistics using the a priori ln P hyperbolicity hypothesis, p(r) Ӎ C/(rγ). Obviously, ≡ ≡ γ γ lnp(r) –s[p(r)] N[p(r)] = lnC – lnr = Sc – S(r); Fig. 4. Statistical identification of (1) the initial and (2) the in other words, either the positive or the negative asymptotic distributions of the quartet alphabet. entropy of the distribution is proportional to the rank entropy. fronts, which, in the global sense, are characterized by Figure 4 shows the result of the statistical identifica- the maximum convexity (in the discrete representation of both initial and asymptotic distributions of the tion). Now, consider some specific features of the con- quartet alphabet. The quality of the verification of the struction of coordination fronts. hyperbolicity hypothesis (Fig. 4) is quite satisfactory, γ γ γ The existence of edge contacts cannot provide the 0 = 0.87, ∞ = 1.05, and = 0.96. Thus, a Penrose continuity of coordination fronts. The number of edge mosaic obeys the ZipfÐMandelbrot statistics. More- α ∨ α ∨ α ∨ contacts does not exceed two. over, P0( 1/v c) = P∞( 1/v c) and P0( 2 /v c) = On average, the number of possible topological ver- P∞(α /v ∨ c) ⇒ {P ; P }, which indicates the 2 gold gold tex contacts between the rhombuses is four, i.e., invariance of these “golden frequencies” also in the exceeds the number of topological edge contacts. problem of eigenvalues of the M-matrix. It is well-known that the mechanism of generation As is seen from Fig. 2, the continuity and the maxi- of hyperbolic statistics in linguistics, economics, and mum convexity of coordination fronts provide vertex sociology is the consequence of the competition contacts with a considerable statistical advantage over assumed to be absent in the collectives with the normal edge ones, (Gaussian) statistics. Figure 2 illustrates the mecha- ()α ∨ α ()α ∨ α nism of hyperbolicity of the quartet g-rhombus alpha- P v / 1 2 ==68%; Pc/ 1 2 32%, bet of a mosaic. Choosing an arbitrary g-rhombus as a percolation center, we can construct the coordination which reflects the golden nature of a Penrose mosaic.

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(a) MAPPING OF A PENROSE MOSAIC INTO QUASISTOCHASTIC CAYLEY’S TREES AND ENTROPY PERCOLATION The conventional coordinate representation of a Penrose mosaic is insufficient because a researcher can reveal only the objective coordinate component (g-rhombuses) of the quasicrystal order. We believe that the essence of cellular net structures and mosaics (including Penrose mosaics) is reflected by another component—a pulse one (see the [2q × 2p]-matrices). The pulse component is understood as the simplest adjacency pair ratio or the coordination between any two contacting pairs of golden rhombuses in a mosaic. Such a representation (adequate for the [2q × 2p] matrix) can be conveniently considered in graph structures of a special form. We mean here the adjacency tree graphs [12] constructed by a certain algorithm. The closest analogues of such graphs are Bethe trees [13] generalized in the form of Cayley’s trees. The tree representation is the most complete one, because it allows one to reveal in explicit form all the coordination relationships without the loss of the objective component [14Ð16]. The algorithm for constructing Cayley’s tree (b) graphs for cellular structures of the general type (such as net structures of mesodefects in quartz glasses) and the properties of Cayley’s trees are described elsewhere [14Ð16]. Here, we should like to emphasize that, in the general case, the degree of branching of the bushes of Cayley’s trees is a random quantity. However, neither reflection in the tree topology nor some order and organization of Cayley’s trees can be forbidden. Cayley’s trees of such type are called quasistochastic. Cayley’s trees of a Penrose mosaic are related to this class. Now, apply the theory of graph enumeration to these Cayley’s supertrees of a Penrose mosaic (Fig. 5) [12, 17]. However, the theory of graph enumeration usually considers decomposition of Cayley’s trees into elementary bush subgraphs ignoring the types of vertices and bonds. In our case, this simplified approach is useless. We introduce a new type of an enumerative polynomial composed not of bushes but of elementary symbols of the [2q × 2p] alphabet. These enumerative Fig. 5. Cayley’s tree of (a) a Penrose mosaic and (b) its polynomials for Cayley’s trees of a Penrose mosaic are enlarged fragment. An obtuse g-rhombus on a Cayley’s tree of a Penrose mosaic is depicted by a circle, an acute arranged according to the ZipfÐMandelbrot statistics g-rhombus, by a triangle. (see the previous Section). Thus, these enumerative polynomials have the fixed rank l = 4. It is natural to normalize this rank enumerative polynomial and, thus, Using the Zipf linguistic interpretation, we see that to obtain the probabilistic enumerative polynomials in an edge contact is more “expensive” (words of higher the form ranks), whereas a vertex contact is “cheaper” and, 4 therefore, is encountered much more often (in our case, l Π ()x = µ()l x , (2) by a factor of 2.13). These opposite tendencies result in i ∑ i an “asymptotic compromise” reflected in the [2q × 2p] l = 1 matrices. Thus, in terms of the ZipfÐMandelbrot law, where x ∈ [2q × 2p]; l = 1, 2, 3, 4 are the ranks of µ a Penrose parquet has typical linguistic characteris- branches in the ZipfÐMandelbrot sense; i(l) are the tics. To some extent, a Penrose mosaic (Fig. 2) can probabilistic weights of the lth rank in the [2q × 2p] be considered as a two-dimensional text of polar alphabet; and i is the number of the hierarchical level on geometry. Cayley’s trees of a Penrose mosaic. Figures 6a and 6c

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(a) (b)

0.55 0.5 0.45 0.4 0.55 0.35 0.5 0.3 0.45 0.25 0.4 Probability 0.2 0.35 0.15 13 0.3 0.1 12 11 0.25 0.05 10 0 9 0.2 Probability Hierarchy8 0.15 13 12 11 7 10 9 8 6 0.1 Hierarchy7 6 5 5 4 2 1 0.05 3 2 3 4 0 4 Rank 3 1 2 3 2 4 Rank (c) (d)

0.55 0.5 0.45 0.55 0.4 0.5 0.35 0.45 0.3 0.4 0.25 0.35 0.2 0.3 Probability 0.15 0.25 0.1 0.2 0.05 Probability 11 0.15 0 9 0.1 13 Hierarchy 0.05 10 11 12 7 8 9 0 6 7 5 1 5 1 2 3 3 4 Hierarchy 3 2 Rank 4 2 1 4 3 Rank Fig. 6. Dynamics of the (a, c) rank probabilistic enumerative polynomials and (b, d) rank probabilistic dynamics over the levels of a Cayley’s tree of a Penrose mosaic along the percolation directions center periphery; (a, b) direct and (c, d) inverse flows. and Figures 6b and 6d show the hierarchical dynamics reverse flow is somewhat smoothened. The essentially of the rank statistics and the partial (rank) dynamics, stochastic nature of the wavelike flow of the probabilis- Eq. (2). One can readily see the obvious periodic nature tic measures is seen in all the representations in Fig. 6, of these polygons, which indicates the quasicrystal which indicates the prevalence of the stochastic compo- nature of a Penrose mosaic. nent over the determined percolation component of Cayley’s trees of a Penrose mosaic. This qualitative Compare qualitatively the hierarchical dynamics of analysis of the polygon percolation shows that it is nec- the rank polynomials (Figs. 6a, 6c) along the percola- essary to convolute this detailed information into a cer- tion directions centerÐperiphery. The alternation of tain functional so that it could reflect both the noiselike decreasing and mode polygons is more clearly seen in and the structurized components in the ordering of Cay- the direct flow (Fig. 6a). The variation of the mode ley’s trees of a Penrose mosaic. structure of polygons is seen in Fig. 6a more clearly than in Fig. 6c. The reflection from the ∞-horizon Now, consider the tree percolation of the probabilis- results in a more homogeneous percolation structure of tic measures of the mosaic at the functional level. In polygons (Fig. 6c). Comparing Figs. 6b and 6d, we see particular, consider the entropy functional set at the that the partial (rank) percolation over the hierarchical probabilistic enumerative polynomial (2) of each levels of Cayley’s trees of a Penrose mosaic in the hierarchy level. The entropy functional was chosen in

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 194 YUDIN et al. η HV old in the problem of vertices or bonds to establish 0.80 2 (a) 18 the relation between this threshold and the lattice 16 symmetry. 0.75 14 The percolation problem considered here is quite 0.70 different in at least two aspects. First, the noiselike fac- 12 0.65 tor is included into the quasistochastic topology of Cay- 10 ley’s trees of a Penrose mosaic (Fig. 6) and, thus, can- 0.60 1 8 not be considered as an external factor. A random quan- 0.55 6 tity in our problem is not only bush branching of Cayley’s trees of a Penrose mosaic, but also the bases 0.50 4 (vertices) of the bushes. It is clear that the internal (b) 20 nature of the stochastics in our case is much more com- 0.80 plicated than in the classical percolation problem. 0.75 18 2 16 Second, percolation in lattices is understood as a 0.70 14 certain generalized flow also of an external nature. The carriers of this flow can be charges, masses, densities, 0.65 12 concentrations, etc., but they all belong to the class of 1 10 0.60 measures. In our case, these are polygons and statistics 8 given by Eq. (2) and, later, also entropy functionals 0.55 6 given by Eqs. (3) and (4). Therefore, one can consider 0.50 4 the probabilistic entropy-like flows in Cayley’s trees of 024681012 a Penrose mosaic. Again, these measures are essentially i internal and, therefore, the problem of probabilistic Fig. 7. (1) Percolation entropy and (2) structural coefficient entropy percolation is a problem of self-identification over the hierarchical levels of Cayley’s tree of a Penrose of the quasistochastic topology of Cayley’s trees of a mosaic: (a) direct (H = 0.6749 ± 4.13%, η = 9.99%, δη = V ± η Penrose mosaic. 37.14%) and (b) reverse (HV = 0.6867 2.66%, = 8.44%, δη = 31.56%) flows. Calculations by Eqs. (3) and (4) for Cayley’s trees of a Penrose mosaic are illustrated in Fig. 7, where the entropy percolation over the hierarchical levels of Cay- Vajda form: ley’s trees of a Penrose mosaic is of the wave type. Esti- H ()i = µ()l []1 Ð µ ()l , (3) mation of the entropy percolation by Eqs. (3) and (4) V ∑ i i (Fig. 7) yields the minimum period consisting, on l average, of two hierarchical levels. According to []µ () Figs. 7a and 7b, the quasiperiodic entropy oscillations η () HV i l V i = 1 Ð ------[]µ ()-. (4) are preserved in both direct and reverse flows, but are sup HV i l more pronounced in the former one. Although the aver- Thus, the entropy introduced here possesses the proper- age values calculated by Eq. (4) in both cases differ ties of nonnegativity, nonlinearity, and connectivity and by 1%, the individual values of these curves can differ obeys the additivity principle. The upper estimate of the by 7%. The pronounced differences in η-values (Fig. 7) asymptotic Vajda entropy equals unity, whereas the indicate the statistical importance of the quasiperiodic- Shannon measure converges. The Vajda entropy corre- ity of the percolation entropy flows in Cayley’s trees of sponds to the chosen scheme with β-distributions, a Penrose mosaic. In terms of thermodynamics, the which is well-known in the theory of Brownian reverse flow is “cooler” than the direct one. One has to motion. Equation (4) can be interpreted as the struc- pay attention to the average entropies given by Eq. (3), tural coefficient of the percolation process in terms of which almost coincide in both cases in Figs. 7a and 7b, whence follows the conclusion that, on average, the µki ⇒ µki η ⇒ entropy. If HV[(i l)] maxHV[(i l)] V(i) 0, the direct and reverse flows in Cayley’s trees of a Penrose entropy of the ith hierarchy is purely stochastic (100% mosaic are equivalent in terms of the entropy func- µki ⇒ η ⇒ tional. The variations in entropy with respect to the noise component). If HV[(i l)] 0, then V(i) 1 (is average value are also essential and are from three to fully determined). Obviously, in real situations, the sto- four percent. Comparing the stochastic and the deter- chastic and the determined percolation components are mined levels in Figs. 7a and 7b, we see that the direct mixed. flow has a 10%-ordered component, whereas the We should like to make an important methodologi- reverse one, an 8.4%-ordered component. Thus, the cal remark on the percolation problem. Usually, one reverse flow (Fig. 7b) is more stochastic than the direct considers percolation of a certain physical “agent” in one, which is the “payment” for reflection from the the lattice whose vertices or bonds are affected by cer- ∞-horizon and can be considered as a certain analogue tain noise factors. We search for the percolation thresh- of the second law of thermodynamics.

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Another no less important result is obtained by com- The (internal) percolation problem for Cayley’s paring the entropy characteristics (see the previous sec- trees of a Penrose mosaic is solved in terms of entropy tion) for a quartet alphabet and the percolation entropy functionals for enumerating rank polynomials. Cay- averaged over the hierarchies of Cayley’s trees of a ley’s trees of a Penrose mosaic are characterized by a Penrose mosaic. In fact, both entropies are practically wave flow in both center directions with a mini- identical. Thus, the percolation of the entropy measures mum period equal to two hierarchical levels of Cayley’s on Cayley’s trees for a Penrose mosaic obeys the ergod- trees of a Penrose mosaic hierarchy. The percolation icity principle extended to Cayley’s tree graphs. More- entropy is characterized by a periodic flow with ~90%- over, on average, the probability and the entropy perco- stochasticity and, thus, with ~10% order. This is an lations on Cayley’s trees of a Penrose mosaic can be important quasicrystal aspect of a Penrose mosaic. On characterized in terms of a certain entropy invariant, average, percolation entropy is equivalent to the golden which corresponds to the golden entropy at the level of entropy of a Penrose mosaic in the [2q × 2p] alphabet, the quartet alphabet. which indicates the ergodicity of the tree percolation. On average, the Markovian shift on Cayley’s trees of a CONCLUSION Penrose mosaic is characterized by a golden-entropy invariant. It is suggested that Penrose mosaics be considered using a quartet alphabet [2q × 2p], whose elements are g-rhombuses (vertex or edge types of contact). In a REFERENCES broader sense, a Penrose mosaic obeys the hierarchical three-level alphabet—a pair of golden triangles and 1. D. Shechtman, I. Blech, D. Gratias, et al., Phys. 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A Penrose mosaic has an invariant (a dou- Kristallografiya 41, 798 (1996) [Crystallogr. Rep. 41, 756 (1996)]. ble ratio at level II) of the probabilities of an asymptotic quartet vector. We also introduced into consideration 7. G. M. Zaslavskiœ, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Usp. Fiz. Nauk 156 (2), 193 (1988) the entropy functionals in alphabets I and II generated [Sov. Phys. Usp. 31, 887 (1988)]. by the golden relationships, so that a Penrose mosaic can be considered as a parquet in the alphabet [2q × 2p] 8. V. V. Yudin, Doctoral Dissertation in Physics and Math- synthesized with the aim of preserving the golden ematics (Vladivostok, 1987). entropy. 9. V. V. Lapatin and Yu. F. Ivanov, Pis’ma Zh. Éksp. Teor. Fiz. 50 (11), 466 (1989) [JETP Lett. 50, 499 (1989)]. It is proven in terms of rank statistics in the alphabet 10. V. A. Malyshev, Usp. Mat. Nauk 53 (2), 107 (1998). [2q × 2p] that, on the whole, ZipfÐMandelbrot distribu- 11. A. M. Bratkovskiœ, Yu. A. Danilov, and G. I. Kuznetsov, tion is valid for a Penrose mosaic. The mechanism of Fiz. Met. Metalloved. 68 (6), 1045 (1989). the generation of hyperbolic statistics is considered as the competition between the expensive edge and the 12. A. A. Zykov, Foundations of Graph Theory (Nauka, cheap vertex contacts in the construction of convex Moscow, 1987). coordination fronts. The results obtained allow the 13. J. M. Ziman, Models of Disorder: the Theoretical Phys- interpretation of a Penrose mosaic on the basis of a lin- ics of Homogeneously Disordered Systems (Cambridge Univ. Press, Cambridge, 1979; Mir, Moscow, 1982). guistic model and, thus, as a linguistic structure or a text with a two-dimensional polar geometry. The com- 14. V. V. Yudin, T. A. Pisarenko, E. A. Lyubchenko, and E. G. Savchuk, Kristallografiya 44, 413 (1999) [Crystal- plete representation of a Penrose mosaic in quasisto- logr. Rep. 44, 373 (1999)]. chastic Cayley’s trees is constructed with the aid of the [2q × 2p] alphabet which, along with the objective com- 15. E. A. Lyubchenko, Candidate’s Dissertation in Physics ponent (g-rhombuses), takes into account the coordina- and Mathematics (Vladivostok, 1999). tion and adjacency of the vertex and edge contacts. 16. T. A. Pisarenko, Candidate’s Dissertation in Physics and Growing and collapsing Cayley’s trees are discussed, Mathematics (Vladivostok, 2000). where the coordination translation is considered as a 17. F. Harary, in Applied Combinatorial Mathematics, Ed. Markovian r-shift. A Cayley’s tree of a Penrose mosaic by E. F. Bechenbach (Wiley, New York, 1964; Mir, Mos- is, in fact, a simplicial complex with ultrametrics. In cow, 1968). terms of crystallography, a Cayley’s tree of a Penrose mosaic is a generalized quasistochastic tree lattice. Translated by L. Man

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Crystallography Reports, Vol. 47, No. 2, 2002, pp. 196Ð200. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 232Ð236. Original Russian Text Copyright © 2002 by Pushcharovskiœ, Pasero, Merlino, Vladykin, Zubkova, Gobechiya. STRUCTURE OF INORGANIC COMPOUNDS

Crystal Structure of Zirconium-Rich Seidozerite D. Yu. Pushcharovskiœ*, M. Pasero**, S. Merlino**, N. V. Vladykin***, N. V. Zubkova*, and E. R. Gobechiya* * Faculty of Geology, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia ** Department of Earth Sciences, University of Pisa, Pisa, 56126 Italy *** Vinogradov Institute of Geochemistry, Siberian Division, Russian Academy of Sciences, ul. Favorskogo 1a, Irkutsk, 664033 Russia Received August 21, 2001

Abstract—The crystal structure of seidozerite was refined (a Siemens P4 diffractometer, MoKα radiation, 1180 independent reflections, anisotropic refinement, R = 0.053). The monoclinic unit-cell parameters are a = 5.627(1) Å, b = 7.134(1) Å, c = 18.590(4) Å, β = 102.68(1)°, sp. gr. P2/c, Z = 4. The structural formula, Na1.6Ca0.275Mn0.425Ti0.575Zr0.925[Si2O7]OF, agrees well with the results of the electron probe analysis. Seidoz- erite is demonstrated to belong to the meroplesiotype polysomatic series including the structures of more than 30 titano- and zirconosilicates. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION EXPERIMENTAL The zirconosilicate seidozerite was discovered by The seidozerite crystals were found in an alkaline E.I. Semenov in 1958 in the Lovozero alkaline massif pegmatitic body 1 m in thickness and 10 m in length [1]. This mineral belongs to a rather large group of located in the southwestern region of the Burpala alkaline massif, whose area and age were evaluated at (Zr,Ti)-silicates, whose structures contain three-layer 150 km2 and 330 Ma, respectively. Seidozerite occurs heteropolyhedral stacks denoted by HOH, where O is as elongated crystals crosscutting earlier catapleiite iso- the central layer consisting of octahedra, and H are the lations and intergrowing with grains of nepheline and outer layers formed by Si-tetrahedra and octahedra of feldspar. highly charged Zr or Ti cations. In 1964, A.M. Portnov According to the results of the electron probe anal- described the second specimen of seidozerite from ysis performed for three crystals, the average chemical North Baikal [2]. The crystal structures of the speci- compositions of the major components of seidozerite mens from both deposits were studied by the photo- are as follows (wt %): SiO2, 29.39; ZrO2, 29.20; TiO2, graphic method [3, 4]. However, the high R factors 13.03; Al2O3, 0.26; MnO, 5.76; CaO, 3.81; FeO, 1.90; (0.166 and 0.207 for the h0l and 0kl reflections, respec- MgO, 0.32; Na2O, 12.26; K2O, 0.05; Nb2O5, 0.68; tively) obtained in the pioneering study [3] and the La2O3, 0.11; Ce2O3, 0.15; and F, 3.06; the sum is 99.98. high, although slightly improved, R factors (0.146 for This composition corresponds to the following chemi- the non-zero 0kl, 1kl, and 2kl reflections and 0.100 for cal formula of seidozerite describing the cation con- the non-zero h0l, h1l, and h2l reflections) obtained in tents (with respect to 36 anions, namely, é2– and F–): the more recent investigation [4] cast some doubt upon (Na Ca K La Ce ) the characteristics of the cation distribution in the 6.29 1.08 0.02 0.01 0.01 seidozerite structure. á (Zr3.77Ti2.59Mn1.29Fe0.42Mg0.13Nb0.08)(Si7.78Al0.08). Recently, seidozerite crystals were also discovered A crystal of dimensions 0.08 × 0.08 × 0.1 mm3 was in another region of the Burpala massif (North Baikal). selected under a microscope and was used for the qual- They are characterized by an higher ZrO2 content itative electron probe analysis (a Philips PW 515 xl 30 (28.18Ð30.17 wt %) compared to those in the speci- microprobe, the accelerating voltage was 20 kV; the mens from the Burpala and Lovozero massifs studied current intensity was 9 nA; and the diameter of the elec- µ carlier (19.6Ð23.15 and 23.14 wt %, respectively). The tron beam was 5 m). The results of the analysis confirmed the presence of the above-mentioned cations in high ZrO content in the new seidozerite specimen, the 2 the mineral. The X-ray diffraction data were collected availability of single crystals suitable for X-ray diffrac- from the same crystal on an automated Siemens P4 dif- tion analysis, and the present-day possibilities of the fractometer. The crystallographic characteristics and instruments have stimulated interest in the refinement the details of the X-ray diffraction study and structure of the seidozerite structure. refinement are given in Table 1.

CRYSTAL STRUCTURE OF ZIRCONIUM-RICH SEIDOZERITE 197

Table 1. Crystallographic characteristics and details of X-ray diffraction study

Formula Na1.6Ca0.275Mn0.425Zr0.925Ti0.575[Si2O7]OF Unit-cell parameters, Å a = 5.627(1), b = 7.134(1), c = 18.590(4) β = 102.68(1)° Sp. gr.; ZP2/c; 4 Unit-cell volume V, Å3 728.06(3) Calculated density ρ, g/cm3 3.574 Absorption coefﬁcient µ, mmÐ1 3.34 Molecular weight 391.8

F000 746.0 Diffractometer Siemens P4 Wavelength, Å 0.71069 Maximum value of the 2θ angle, deg 60.01 Total number of reflections 3291 Total number of independent reflections 2122 | | σ Number of independent reflections with F > 4 (F) 1180 Rint, % 7.81 Number of parameters in the refinement 138

RF upon isotropic refinement 0.081 RF upon anisotropic refinement 0.053 wR(F2) 0.145 GOF 0.990 ∆ρ 3 max, e/Å 1.09 ∆ρ 3 min, e/Å Ð1.27

Table 2. Distribution of the cations over the positions in the seidozerite structure Electron Content Content Average Posi- Electron content taking Sum of according according The present study cationÐanion tion content into account the ionic radii to study [3] to study [4] distance real occupancy

Zr Zr0.75Ti0.25 Zr0.7(Mn, Fe)0.3 Zr0.8Ti0.2 36.24 36.40 2.127 2.057 Ti Ti Ti0.39Fe0.11 Ti0.35Zr0.125 24.86 25.40 1.994 1.928 Mn Mn(Mg) Ca0.42Mn0.08 Mn0.29Na0.125Ti0.025 18.35 18.35 2.265 2.153

Na(1) Na Na0.78Ca0.22 Na0.965Ca0.035 11 11.31 2.568 2.540 Na(2) Na Na Na0.43Ca0.07 12.28 12.28 2.498 2.534 Ca Na Na Ca0.170Mn0.135Na0.080 15.28 15.31 2.394 2.300

Note: The cation content in the unit cell is Na6.3Ca1.1(Mn, Fe)1.7Zr3.8Ti2.6 according to the data of the electron probe analysis and Na6.4Ca1.1(Mn, Fe)1.7Zr3.7Ti2.3 , according to the results of X-ray diffraction study.

The unit-cell parameters were determined by the subsequent stages, the electron contents of six cationic least-squares method based on the angular parameters positions were refined, and the anisotropic thermal of 40 reflections in the range 8° ≤ 2θ ≤ 27°. The absorp- parameters were included, which made it possible to tion correction was applied using the ψ scanning proce- reduce the R(F) factor to 0.053. The distribution of the dure. The structure was refined within the space cations over six nonequivalent positions was estab- group P2/c starting from the atomic coordinates deter- lished based on the refinement of their electron con- mined earlier [4]. Initially, the structure was refined iso- tents, the correspondence between the structural for- tropically to R = 0.081 for 1180 reflections with |F| > mula and the data of electron probe analysis, the elec- 4σ(F) using the SHELX97 program package [5]. At troneutrality of the chemical formula, the valence

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 198 PUSHCHAROVSKIŒ et al.

Table 3. Coordinates, thermal parameters, multiplicities (Q), and occupancies (q) for the basis atoms

** 2 × Atom* x/ay/bz/cQqUeq , Å 100 Zr 0.2004(1) 0.1193 (1) 0.07342(4) 1 1 0.97(2) Ti 0.0 0.1153(3) 0.25 0.5 0.475 1.14(3) Mn 0.5 0.3566(4) 0.25 0.5 0.440 1.74(6) Na(1) 0.2051(6) 0.6157(5) 0.0695(1) 1 1 1.13(5) Na(2) 0.0 0.6149(8) 0.25 0.5 0.5 2.18(9) Ca 0.5 0.8656(7) 0.25 0.5 0.385 3.32(8) Si(1) 0.7273(4) 0.3872(4) 0.1046(1) 1 1 1.00(4) Si(2) 0.7217(4) 0.8422(3) 0.1049(1) 1 1 0.99(4) O(1) 0.738(1) 0.6135(8) 0.1096(3) 1 1 2.3(1) O(2) 0.451(1) 0.3264(9) 0.0680(3) 1 1 2.1(1) O(3) 0.440(1) 0.9015(9) 0.0763(3) 1 1 2.0(1) O(4) 0.927(1) 0.3267(9) 0.0586(3) 1 1 1.6(1) O(5) 0.904(1) 0.9073(8) 0.0518(3) 1 1 1.4(1) O(6) 0.797(1) 0.3154(8) 0.1900(3) 1 1 1.6(1) O(7) 0.817(1) 0.9141(9) 0.1888(3) 1 1 1.7(1) O(8) 0.240(1) 0.124(1) 0.1849(3) 1 1 1.9(1) F 0.301(1) 0.600(1) 0.1922(3) 1 1 3.9(2) * The cation contents of the positions correspond to those presented in Table 2. ** The Ueq values are calculated based on anisotropic thermal displacements of the atoms. balance, and the fact that the average interatomic dis- ing polyhedra. The resulting distribution (Table 2) dif- tances should be approximately equal to the sum of the fers from the distributions found earlier [3, 4] and is ionic radii of the cations and anions in the correspond- supported by the lowest R factor. The structural formula 2(Na,Ca){(Na,Ca)(Ca,Mn,Na,ᮀ)(Mn,Na,Ti,ᮀ)(Ti,Zr,ᮀ) Table 4. Interatomic distances (Å) [(Zr,Ti)Si2O7]OF} (the content of the H layer is enclosed in brackets, and the content of the HOH stack ZrÐO(8) 2.035(5) Na(1)ÐF 2.229(6) CaÐF 2.340(8) is enclosed in braces; Z = 4) agrees well with the results ÐO(3) 2.051(6) ÐO(4) 2.370(6) ÐF' 2.340(8) of electron probe analysis. The distribution of the cat- ÐO(2) 2.061(6) ÐO(3) 2.419(7) ÐO(7) 2.345(6) ions presented in Table 2 should be considered as the ÐO(4)' 2.110(6) ÐO(2) 2.488(7) ÐO(7)' 2.345(6) optimum compromise between the experimental results ÐO(5) 2.223(6) ÐO(4)' 2.570(7) ÐO(8) 2.497(7) of chemical and X-ray analysis. Conceivably, there are insignificant variations in the cation contents in partic- ÐO(5)' 2.281(5) ÐO(5)' 2.659(7) ÐO(8)' 2.497(7) ular polyhedra, which are, apparently, responsible for Average, 2.127 ÐO(1) 2.886(7) Average, 2.394 the higher values of Ueq for the cations located in the Ca ÐO(1)' 2.925(7) position compared to Ueq for the Na(1) and Na(2) posi- MnÐF 2.209(8) Average, 2.568 TiÐO(7) 1.974(6) tions. ÐF' 2.209(8) ÐO(7)' 1.974(6) The final coordinates of the basis atoms and their ÐO(6) 2.225(6) Na(2)ÐF 2.200(5) ÐO(8) 2.002(5) thermal parameters are given in Table 3. The inter- ÐO(6)' 2.225(6) ÐF' 2.200(5) ÐO(8)' 2.002(5) atomic distances are listed in Table 4. The valence bal- ÐO(8) 2.361(7) ÐO(7) 2.529(8) ÐO(6) 2.007(6) ance was calculated according to Brese and O’Keeffe ÐO(8)' 2.361(7) ÐO(7)' 2.529(8) ÐO(6)' 2.007(6) [6], and the results are presented in Table 5. The Si(1)Ð Average, 2.265 ÐO(6) 2.560(8) Average, 1.994 O and Si(2)ÐO bond lengths are (1.610(6)Ð1.632(6) ÐO(6)' 2.560(8) and 1.612(6)–1.637(6) Å, respectively with average ÐO(1) 2.702(6) values 1.618 and 1.626 Å, respectively) close to the standard values and are therefore not given in Table 4. ÐO(1)' 2.702(6) The projection of the structure (the ATOMS program [7]) Average, 2.498 is shown in the figure.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 CRYSTAL STRUCTURE OF ZIRCONIUM-RICH SEIDOZERITE 199

RESULTS AND DISCUSSION Na(1) Si The crystal structure of seidozerite is based on the C HOH bafertisite-like stacks parallel to the (001) plane Ti [8]. The O octahedral layer is composed of (Ti,Zr)O6-, (Mn,Na,Ti)O -, and (Ca,Mn,Na)-octahedra and the 6 Ca eight-vertex Na(2)-polyhedra. All the cationic positions within the octahedra of the O layer are only partly occu- Mn pied.

On the contrary, the (Zr,Ti)O6-octahedra, which are Na(2) involved in the heteropolyhedral H layers together with the Si2O7 diortho groups, are completely occupied. The HOH stacks in seidozerite are directly linked to each other via edges shared by the adjacent (Zr,Ti)O6-octahedra belonging to different stacks. Hence, the seidozerite structure can be described as a mixed framework, whose cavities located between the adjacent HOH Zr stacks are occupied by the Na(1) cations occupied in the eight-vertex polyhedra. The valence balance cannot be calculated with a high degree of accuracy because of the above-mentioned cationic disordering. However, the results presented in Table 5 allow the unambiguous identification of the positions occupied by the F atoms or the OH groups. The O anions in the O(8) position and the F b anions belong to the O layer, but they are not involved in the Si-tetrahedra. The modular structures of more than 20 titanosilicates containing the bafertisite-like HOH stacks analo- Seidozerite structure projected onto the (100) plane. gous to those found in seidozerite have been considered earlier [8Ð11]. The crystal structures of some of these minerals are still unknown, and they were assigned to All titanosilicates of this series have close unit-cell the bafertisite-like group based on the comparison of parameters (a ~ 5.5 Å, b ~ 7 Å) typical of the bafer- the unit-cell parameters and the data on their compositions. The correctness of the application of the modular tisite-like HOH stacks. The space between these stacks, concept to the prediction of unknown structures has like that in layered silicates, can involve isolated atoms already been confirmed in the studies of minerals such or atomic groups as well as the fragments of other min- as delindeite, perraultite, betalomonsovite, and some eral types. The latter statement is exemplified by the varieties of lamprophyllite, whose structures were structures of quadruphite and polyphite [12], which established more recently. These structures were sur- contain nacaphite [13] fragments between the layers. veyed in most detail in [11]. The interlayer content (according to Belov’s evocative

Table 5. Calculated valence balance Anion, cation O(1) O(2) O(3) O(4) O(5) O(6) O(7) O(8) F Zr 0.675 0.694 0.592 0.436 0.724 0.373 Ti 0.624 0.682 0.632 Mn 0.274 0.190 0.220 Na(1) 0.053 0.156 0.188 0.214 0.098 0.225 0.048 0.125 Na(2) 0.094 0.138 0.151 0.262 Ca 0.229 0.152 0.170 Si(1) 1.016 1.030 1.039 0.979 Si(2) 0.968 1.033 0.965 1.019 Σ 2.179 1.861 1.915 1.970 1.872 2.015 2.081 1.698 0.877

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 200 PUSHCHAROVSKIŒ et al. expression, the “pie filling”) determines the c parame- the program “Russian Universities.” S. Merlino and ter, which increases as the filling becomes more com- M. Pasero acknowledge the support of the MURST plex. The thickness of the HOH stacks is ~9–10 Å. In project (“Structural Imperfections in Minerals: Micro- the structures where the interlayer space is occupied structure, Modular Aspects, and Structure Modula- only by isolated atoms, the c-parameter is ~n × 10 Å, tion”). D.Yu. Pushcharovskiœ acknowledges the support where n is the number of the HOH stacks. The c param- of the Russian–Italian Scientific Cooperation program eter in seidozerite (18.590 Å) indicates that its unit cell (project no. 62). contains two symmetrically equivalent HOH stacks perpendicular to the [001] axis. In the seidozerite structure, the HOH stacks are linked via the shared edges of REFERENCES the Zr-octahedra, and d (0 0 1)/n has the minimum 1. E. I. Semenov, M. E. Kazakova, and V. I. Simonov, Zap. value (9.068 Å) of all minerals of this series. In the Vseross. Mineral. OÐva. 87 (5), 590 (1958). polyphite structure characterized by a large interlayer 2. A. M. Portnov, Dokl. Akad. Nauk SSSR 156 (2), 338 space, d (0 0 1)/n, on the contrary, reaches 26.49 Å. (1964). Within the framework of the classification of homolo- 3. V. I. Simonov and N. V. Belov, Kristallografiya 4 (2), 163 gous and polysomatic series proposed recently [14], (1959) [Sov. Phys. Crystallogr. 4, 146 (1960)]. seidozerite, like all bafertisite-like minerals, belongs to 4. S. M. Skshat and V. I. Simonov, Kristallografiya 10 (5), the meroplesiotype series. This series is referred to as a 591 (1965) [Sov. Phys. Crystallogr. 10, 505 (1965)]. merotype series because one fragment, namely, the 5. G. M. Sheldrick, SHELX97: Program for the Solution HOH stack, is common to all the members of this and Refinement of Crystal Structures (Madison, 1997). series, whereas the second fragment located between 6. N. E. Brese and M. O’Keeffe, Acta Crystallogr., Sect. B: the layers is individual for each mineral of this series. Struct. Sci. B47, 192 (1991). At the same time, the bafertisite-like polysomatic series 7. E. Dowty, Atoms 3.2: A Computer Program for Display- should be considered as a plesiotype series because the ing Atomic Structures (Kingsport, TN 37663, 1995). HOH stacks differ from each other not only in compo- 8. G. Ferraris, Modular Aspects of Minerals / EMU Notes sition but also in topology. Earlier, it has been estab- in Mineralogy 1, 275 (1997). lished that these structural differences can be seen from 9. Yu. K. Egorov-Tismenko, Kristallografiya 43, 306 (1998) the fact that the “octahedral” cations within the H layers [Crystallogr. Rep. 43, 271 (1998)]. can adopt different coordination numbers (either 6 or 5). 10. C. C. Christiansen, E. Makovicky, and O. N. Johnsen, Similar to delindeite studied recently, seidozerite exem- Neues Jahrb. Mineral., Abh. 175, 153 (1999). plifies the plesiotype character of the HOH layers associated with the change in the coordination of the cations 11. G. Ferraris, G. Ivaldi, D. Yu. Pushcharovsky, et al., Can. not only in the H layers but also in the O layers. Thus, Mineral. 39 (5), 1306 (2001). the Na cations involved in the O layers in the delindeite 12. A. P. Khomyakov, G. N. Nechelyustov, E. V. Sokolova, and seidozerite structures are located in seven- and and G. I. Dorokhova, Zap. Vseross. Mineral. OÐva. 121 (3), eight-vertex polyhedra, respectively, rather than in 105 (1992). octahedra. 13. E. V. Sokolova, Yu. K. Egorov-Tismenko, and A. P. Khomyakov, Dokl. Akad. Nauk SSSR 304 (3), 610 (1989) [Sov. Phys. Dokl. 34, 9 (1989)]. ACKNOWLEDGMENTS 14. E. Makovicky, Modular Aspects of Minerals/EMU This study was supported by the Russian Founda- Notes in Mineralogy 1, 315 (1997). tion for Basic Research (project nos. 00-05-65399, 00-15-96633, and 01-05-97243-r2001, Baikal) and by Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 201Ð212. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 237Ð248. Original Russian Text Copyright © 2002 by Sobolev, Golubev, Krivandina, Marychev, Chuprunov, Alcobe, Gali, Pascual, Rojas, Herrero.

STRUCTURE OF INORGANIC COMPOUNDS

Ba1 Ð xRxF2 + x Phases (R = GdÐLu) with Distorted Fluorite-type Structure—Products of Crystallization of Incongruent Melts in the BaF2ÐRF3 Systems. I. Ba0.75R0.25F2.25 Crystals (Synthesis and Some Characteristics) B. P. Sobolev*, A. M. Golubev**, E. A. Krivandina*, M. O. Marychev***, E. V. Chuprunov***, X. Alcobe****, S. Gali****, L. Pascual*****, R.-M. Rojas*****, and P. Herrero***** * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia e-mail: [email protected] ** Bauman State Technical University, Vtoraya Baumanskaya ul. 5, Moscow, 107005 Russia *** Nizhni Novgorod State University, pr. Gagarina 23, Nizhni Novgorod, 630600 Russia **** Universidad de Barcelona, Barcelona, Spain ***** Instituto de Ciencia de Materiales, Madrid, Spain Received June 28, 2001

Abstract—The transparent monolithic crystal rods grown by crystallization of the incongruent Ba0.75R0.25F2.25 melts (R = GdÐYb) are shown to be heterogeneous: the larger part of the crystal volume has a distorted ﬂuorite- type cubic lattice, while the smaller part retains an undistorted cubic structure. The distortions were recorded by the methods of X-ray powder and electron diffraction and crystal optics. Two types of chemical heterogeneity of Ba0.75R0.25F2.25 crystals are considered as possible sources of distortions: (1) a macroheterogeneous (at a level of 1 mm) distribution of RF3 over the crystal-rod diameter (a cellular substructure) and (2) a microheterogeneous (cluster) structure with the nanometer dispersion typical of nanostructured materials. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION structure. The phase diagrams and the composition studied by the X-ray diffraction method are reviewed in The M1 – xRxF2 + x phases (M = Ca, Sr, Ba, Cd, and [1, 2] and described in a number of original publica- Pb; R is one of 16 rare earth elements) with the defect tions. CaF -type structure are characterized by pronounced 2 It is shown that the equilibrium M R F phases nonstoichiometry (0 < x < 0.5). The changes in the crys- 1 – x x 2 + x tal composition give rise to changes in the “defect are cubic, but the crystals of the same composition structure” and crystal properties over a wide range. The grown from melt under nonequilibrium conditions are M R F crystals are multicomponent materials that not necessarily cubic. 1 – x x 2 + x The second source of information on the system can substitute single-component crystals (CaF2, BaF2, etc.), whose commercial application is very limited. (cubic) of the M1 – xRxF2 + x phases is the study of their structures performed mainly on nonequilibrium sam- It is commonly believed that all the M1 – xRxF2 + x ples. The standard methods fail to reveal the distortions phases are crystallized from melts as cubic crystals, sp. of the fluorite unit cell, which are seen from the split- gr. Fm3m with the fluorite-type (defect) structure. ting of reflections. However, what are the reasons to believe it and how jus- Thus, the structural studies of M1 – xRxF2 + x crystals tified is this opinion? grown from melts cannot show that they are cubic This opinion is based on two types of data. First, either. these are the data obtained from the study of the phase The only fact that can be stated is that crystallization from a melt provides the formation of the most pro- diagrams of the MF2–RF3 systems. To study the phase equilibria in the subsolidus, a mixture of powdered nounced disordered phases, i.e., nonstoichiometric flu- components is annealed until the attainment of the orite cubic M1 – xRxF2 + x phases. equilibrium state. Then the samples are quenched and Thus, the opinion that the M1 – xRxF2 + x phases crys- their phase composition is studied by the X-ray diffrac- tallized from melts are cubic fluorite-type phases fol- tion method. At rather high temperatures, numerous lows more from general considerations and traditional M1 – xRxF2 + x phases showed no distortions in cubic concepts than from experimental facts.

202 SOBOLEV et al.

The history of the studies of M1 – xRxF2 + x crystals SAMPLES includes a case where “obvious” and generally In the MF2ÐRF3 systems (M = Ca, Sr, Ba, Cd, and Pb; accepted concepts were disproved several years later. R is a rare-earth element), we revealed 80 phases of the We mean here the widespread recognition of the theo- composition M R F . These phases are divided retical model suggested by Goldschmidt to describe the 1 – x x 2 + x into five families depending on the number of MF2-cat- isomorphism of Ca2+ and R3+ in CaF [3]. According to 2 ions. If the effect of the heterogeneity of M1 – xRxF2 + x this model, the superstoichiometric F1–-ions occupy the crystals is seen in the X-ray powder diffraction pat- centers of large cubic voids. The logics and the apparent terns, they should be most pronounced for the phases of uniqueness of this model (there are no other large voids) the Ba1 – xRxF2 + x family. We studied the phases with allowed the model to survive from 1926 to 1969, when, R = GdÐLu possessing the maximum difference in M2+ finally, experimental structural studies of Ca0.61Ce0.39F2.39 3+ and R sizes (among all M1 − xRxF2 + x phases) and a [4] and Ca0.9Y0.1F2.1 [5] crystals were undertaken and pronounced dependence of the lattice parameter on the proved that the model was erroneous. composition, which is a favorable condition for the observation of the distortions of a cubic lattice. The story of the isomorphism model is analogous to the story discussed in the present paper and both stories The Ba0.75R0.25F2.25 compositions were chosen from are closely related. Beginning with the structural stud- the homogeneity region of the Ba1 – xRxF2 + x phases because such an isoconcentration “cut” of the RE series ies of 1969Ð1970, Ca1 – xRxF2 + x crystals were considered to be cubic, which was never verified experi- provides an opportunity to study a new type of cation mentally not only for these crystals but also for other ordering discovered in Ba0.8Yb0.2F2.2 [6] along with MF -based phases. However, as we have already phases containing other rare-earth elements. The com- 2 2+ showed, this statement is true only for the equilibrium plete ordering of Ba and rare-earth elements, accom- compositions. panied by a change in the sp. gr. Fm3m to sp. gr.

In fact, the growth of M1 – xRxF2 + x crystals is a non- Pm3m is attained for the composition Ba0.75R0.25F2.25. equilibrium process. The M1 − xRxF2 + x phases for most We selected the samples in two ways. The samples of x values are melted incongruently. The growth of obtained from the mixture of small pieces taken from crystals of these phases is accompanied by the forma- different portions of the crystal rod allowed us to reveal tion of two types of macroheterogeneities in the R3+ dis- the maximum fluctuations in the composition, because tribution—axial and radial (the so-called cellular sub- all types of inhomogeneities (axial and radial) and, structure). Generally, the composition changes mono- hence, all types of the distortions of the cubic lattice in tonically along the growth direction of the crystalline the crystal bulk were apparent. rod, being a background against which the radial inho- The samples, obtained by grinding monolithic crys- mogeneity is developed. tal blocks, allowed us to extract information on the local changes in the composition and the related distor- In M1 – xRxF2 + x crystals, the elements of the sub- tions of the crystal lattice. Hereafter, these samples are structure (the cells and their boundaries) have a milli- called monoblocks and denoted as M. The monoblock meter scale. The variations in their composition give dimension along the growth axis (typically, 2Ð4 mm) rise to lattice-parameter gradients resulting in mechan- did not exceed ~10% of the rod length, and, therefore, ical stresses and, in turn, in the optical anisotropy of the the axial inhomogeneity could be neglected. crystals. The optical anisotropy occasionally observed in M1 – xRxF2 + x crystal rods has not been studied and EXPERIMENTAL was attributed to residual thermal stresses that could not lead to considerable lattice distortions. The crystals were grown by the Bridgman method in an apparatus with a graphite heater. The BaF2 charge On the scale of crystal structures, fluorite-type consisted of the pieces of single crystals containing M1 – xRxF2 + x phases should be considered as microinho- about 0.012 wt % of oxygen. The RF3 reagents had a mogeneous. Superclusters of defects and their associ- purity of 99.90Ð99.99%; they were purified of oxygen ates with nanometer linear dimensions are enriched by the method described in [7] by keeping the melts with R3+, whereas their composition is essentially dif- overheated by 100Ð150°C in a fluorinating atmosphere ferent from the MF matrix of these clusters. of the products of polytetrafluoroethylene (Teflon) 2 pyrolysis for 2Ð4 h up to the attainment of 0.005Ð The aim of the present work is to establish whether 0.060 wt % of oxygen. the inhomogeneities of the chemical composition of the A graphite crucible with up to seven cells filled with products formed in the nonequilibrium crystallization the charge was placed into the growth chamber first Ð2 of incongruent Ba0.75R0.25F2.25 melts (where R = Gd, Tb, evacuated to 10 mm Hg and then filled with He. The Dy, Er, Tm, Yb, and Lu) affect the structural and crys- fluorinating atmosphere was created by the products of tallooptical characteristics of these crystals. Teflon pyrolysis. The velocity of the crucible lowering

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 Ba1 Ð xRxF2 + x PHASES 203

Fig. 1. The shape of Ba1 Ð xRxF2 + x crystals. was 4.9 ± 0.8 mm/h. The cooling rate upon crystalliza- chemical composition was estimated using the parame- ° tion was 100Ð150 C/h. The crystal rods were 35Ð ter of the pseudocubic unit cell apfl calculated using 40 mm in length and 10Ð12 mm in diameter. For the unsplit fluorite reflections. This seems to be admissible spectroscopic studies, the crystals were doped with because the change in the molar volumes of the cubic 3+ 3+ 3+ 3+ 0.01 wt % of Ce , Nd , Pr , or Er . In the text, the M1 – xRxF2 + x phases and the molar volumes of ordered concentrations of these rare-earth elements are indi- phases lie practically on the same straight line [9]. cated together with the concentration of the main RE If the distortions of the cubic fluorite-type lattice element up to an RF3 concentration of 25 mol %. In the were essential, we used the parameter equal to the cube figures, the true compositions are indicated. root of the volume of the distorted unit cell, which is, of The crystals were annealed either wrapped in Ni course, a very rough estimate. (The compositions are foil or in capillaries placed into a sealed Ni-container. given in mol % or, in formulas, in molar fractions (x) The fluorinating atmosphere was created by the prod- of RF3.) ucts of polytetrafluoroethylene pyrolysis. We showed The phase composition of the crystals was studied that this technique provides conditions that practically on the following X-ray powder diffractometers: Sie- exclude pyrohydrolysis. The crystals were annealed for mens D-500 (CuKα -radiation), INEL (CuKα -radia- 14 days at 900°C and for 32 days at 750°C. In the first 12, 1 tion), and HZG-4 (CuKα -radiation). The experiments case, the container was quenched by cooling at a rate of 12, 200Ð300°C/min from the annealing temperature to on an INEL diffractometer were performed on small 400Ð500°C; in the second case, at a cooling rate of samples (~0.1 mm in diameter and 3Ð4 mm in height); 200Ð300°C/s. under certain conditions, this can yield essentially dif- The cellular substructure of the crystals was stud- ferent results from those obtained on other diffractome- ied in the transmitted light of an HeÐNe laser and in the ters where 0.1Ð0.3 g of the material is used. The crys- polarized light in a MIN-8 microscope. The samples for tals were grinded under acetone in an agate mortar. The these observations (disks with a thickness of ~2 mm) information on reflection broadening and splitting was were cut out from the crystal rods normally to the obtained from profile analysis using the PeakFit.v4 pro- growth axis and then were polished (Fig. 1). gram and the approximating Pearson VII function. The chemical composition of the crystals was estimated by the dependence of the unit-cell parameters of EXPERIMENTAL RESULTS Ba1 – xRxF2 + x on the RF3 content [8] obtained for cubic The X-ray diffraction study of as grown phases brought into equilibrium by annealing. For the Ba1 − xRxF2 + x crystals (R = GdÐLu) revealed strong com- Ba1 – xRxF2 + x crystals (where the weak distortions of the position-dependent distortions of the fluorite lattice. CaF2-type lattice are not removed by annealing), the Below, we qualitatively compare diffraction patterns

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 204 SOBOLEV et al.

I, 103 pulse/s (a) (422) (331) Ba0.75 Gd0.25 F2.25 (311) Indices annealed for 14 days in the ﬂuorite-type cell, CuKα 1 at 900°C 4 Ba0.75 Gd0.249 Er0.001F2.25 as grown

(420) 3 (333; 511)

1 70 75 80 (b) (220) Ba0.75 Er0.249 Nd0.001F2.25 (311) annealed for 32 days at 750°C 6

(222) (331) 4 Ba0.75 Er0.249 Nd0.001F2.25 as grown; indices in the ﬂuorite-type cell , CuKα1 (422) (420) 2 (333) (400)

45 50 55 6065 70 75 80 2θ, deg

Fig. 2. A fragment of the X-ray diffraction patterns from (a) an as grown Ba Gd F crystal and the same crystal annealed ° 0.75 0.25 2.25 ° for 14 days at 900 C and (b) an as grown Ba0.75Er0.25F2.25 crystal and the same crystal annealed for 32 days at 750 C. from distorted and undistorted (cubic) samples. We also pseudofluorite unit cell determined from single confirmed the sample distortions by their study in unbroadened 220 and 222 reflections is apfl = 6.0632 Å polarized light. The determination of the phase compo- (This corresponds to 24.05% GdF3.) sition and indexing of the X-ray diffraction patterns are the subject of a separate study. It is evident that, upon annealing, the 420 and 422 reflections are not split any more. However, profile The X-ray diffraction patterns of as grown analysis shows that some earlier unbroadened reflections Ba0.75Gd0.25F2.25 crystals have broadened and split become broadened upon annealing. Fourteen-day anneal- “fluorite” reflections (Fig. 2a). Diffraction patterns of ing at 900°C only partly removed crystal distortions. as grown and annealed crystals are shown by solid The parameter apfl (this corresponds to 25.1% GdF3) lines; the diffraction patterns of the same crystals upon determined with the use of the 331 reflection is apfl = annealing, by dashed lines. The unit-cell parameter of a 6.0573 Å.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 Ba1 Ð xRxF2 + x PHASES 205

I, 103 pulse/s (111) (a)

Indices in the Ba0.75 Tm0.249 Er0.001F2.25 ﬂuorite-type unit cell , CuKα1 as grown 6 (220)

(331) (311) 4 (200) (420) (620) 2 (400) (222) (422) (531) (333) (440) (442)

(331) (b) 8 (200) (111) Indices in the Ba0.75 Yb0.249 Er0.001F2.25 ﬂuorite-type cell , CuKα1 as grown

4 (220) (311) (422) (333) (531) (420) 2 (222) (400) (620) (442)

30 40 50 6070 80 90 100 2θ, deg

Fig. 3. X-ray diffraction patterns from (a) an as grown Ba0.75Tm0.25F2.25 crystal and (b) an as grown Ba0.75Yb0.25F2.25 crystal.

It is well known that the attainment of equilibrium unsplit and unbroadened 440 and 422 reflections is during the annealing of crystals grown from a melt apfl = 6.0304 Å. occurs more slowly than in crystals obtained by solid The Ba Dy F crystals showed minor distor- phase synthesis from components. The kinetics of the 0.75 0.25 2.25 tions; the 400, 331, 440, 531, and 422 reflections are transition of as grown crystals into equilibrium cubic broadened. The unit-cell parameter a (27.1% DyF ) phases at high temperatures should be studied sepa- pfl 3 determined from the broadened 620 reflection is apfl = rately. 6.0292 Å, while those determined from unbroadened 333 and 422 reflections are a = 6.0498 Å (23.9%) and The Ba0.75Tb0.25F2.25 crystals are less distorted than pfl apfl = 6.0269 Å (27.5%), respectively. The considerable the Ba0.75Gd0.25F2.25 ones. Only the fluorite reflection, 400, is split; the 311, 331, 422, and some other reflec- difference in the apfl values determined from different tions are broadened. The 220, 222, 420, 333 + 511, and reflections (see Discussion) does not allow their averaging. 442 reflections are unsplit and unbroadened. The unit- The Ba0.75Er0.25F2.25 crystals grown from a melt have cell parameter apfl (27.3% TbF3) determined from a considerably distorted fluorite-type lattice (Fig. 2b).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 206 SOBOLEV et al.

I, 103 pulse/s

(111) (220) (311) 9 Ba0.75 Lu0.25 F2.25 + Ba0.74 Lu0.26 F2.26 synthesized for 14 days from the components at 905°C; indices in the ﬂuorite-type cell , CuKα1

Ba0.75 Lu0.249 Er0.001F2.25 (222) 6 (200) as grown; indices in the ﬂuorite-type cell , CuKα1 (331)

(400) (422) (420) 3 (110)

30 40 50 60 70 2θ, deg

Fig. 4. Comparison of the X-ray diffraction patterns from (a) an as grown Ba0.75Lu0.25F2.25 crystal and (b) the mixture of the Ba0.75Lu0.25F2.25 and Ba0.74Lu0.26F2.26 phases (solid-phase synthesis).

Assuming the existence of two cubic phases with dif- No monolithic crystal rod with the composition ferent unit-cell parameters in as grown Ba0.75Er0.25F2.25 Ba0.75Lu0.25F2.25, is formed—it decomposes into small crystals is inconsistent with the experimental fact—the 1- to 3-mm-large pieces, which are reminiscent of the presence in the diffraction pattern of unsplit reﬂections result of a polymorphic transition. Thus, in this such as broadened 400 (Fig. 2b) and unbroadened 620 instance, samples were prepared from a mixture of reﬂections (24.3% ErF3), which yields apfl = 6.0315 Å. small pieces of different parts of the rod.

The Ba0.75Tm0.25F2.25 crystals are characterized by Figure 4 shows X-ray diffraction patterns of as the most pronounced distortions. Fig. 3a shows the grown Ba0.75Lu0.25F2.25 crystals and the mixture of the X-ray diffraction pattern (obtained on an INEL diffrac- phases of close compositions obtained by solid phase tometer) from the mixture cut out from different parts synthesis. The diffraction pattern from the sample of an of the rod, which shows all the types of inhomogene- as grown crystal showed some distorted fluorite reflec- ities and distortions. No superstructure reflections are tions. At small angles, a weak reflection forbidden by θ ° observed at small angles (with 2 up to 2 ). Here, the sp. gr. Fm3m is formed. The simulation of the X-ray suggestion that there is a mixture of cubic phases with powder diffraction pattern under the assumption that all different parameters is ruled out by the presence of 2+ unsplit reflections. The weakly broadened 400 and 620 Ba ions are located in the face centers (100% occupancy) shows a possible transition with the change of reflections (25.7% TmF3) yield apfl = 6.0174 Å. the symmetry Fm3m Pm3m with the preserva- The Ba Yb F crystals show strong lattice 0.75 0.25 2.25 tion of the lattice parameter and appearance of weak distortions. The X-ray diffraction pattern (an INEL dif- reflections, one of which corresponds to the experimen- fractometer) from the powder prepared from a mixture tally observed 110 reflection. of small transparent pieces of the crystal rod is shown in Fig. 3b. At the angles 2θ < 20°, no superstructure The 14-day-long solid phase synthesis from the reflections were observed, whereas outside this region components at 905°C yielded a cubic fluorite-type there were superstructure reflections. The suggestion phase (Fig. 4). The X-ray powder diffraction pattern that there is a mixture of several cubic phases is incon- was obtained from the mixture of Ba0.75Lu0.25F2.25 and sistent with the presence of unsplit reflections. Ba0.74Lu0.26F2.26 in approximately equal quantities. At Using the 333 + 511 reflection, we obtain apfl = 6.0137 Å larger angles, a doubling of reflections was observed. (25.5% YbF3). The use of the 620 reflection yielded the phase compo-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 Ba1 Ð xRxF2 + x PHASES 207 sitions 25.1 and 25.9% LuF3, i.e., the compositions cubic; their diffraction patterns show superstructure close to nominal compositions, which demonstrates a reflections and increased unit-cell parameters. high resolution of the diffractometer with a difference The (rhα') Ba δR δF δ phases are described in in the composition of only 1 mol %. 8 + 6 – 34 – [12, 13] as the phases with the composition 7BaF2 á 5TmF3 (41.7 mol % TmF3). These are found for R = DISCUSSION Sm, GdÐLu, and Y in [14] and attributed the composition Ba4R3F17 (42.86 mol % RF3). Similar phases with The composition fluctuations in the Ba0.75R0.25F2.25 R = CeÐNd and Eu were obtained in [15, 16]. The struc- rods containing rare-earth elements from the second ture of Ba4Y3F17 and Ba4Yb3F17 crystals was studied half of the series under consideration lead to the distor- in [17]. tions in fluorite-type cubic lattices revealed from the Thus, in the BaF2–RF3 systems (R = GdÐLu), two corresponding X-ray powder diffraction patterns. types of fluorite-like phases were observed in the com- Because the BaF –RF systems are characterized by 2 3 position range 0Ð50 mol % RF3—cubic phases with the ordering processes with the formation of phases with “fluorite” (small) parameter and ordered phases with structures derived from the fluorite type, we compare increased parameters and superstructure reflections on their X-ray diffraction characteristics with those of the diffraction patterns. The X-ray diffraction patterns Ba0.75R0.25F2.25. for the typical representatives of these phases are shown in Figs. 5a and 5b. The cubic Ba0.75Lu0.25F2.25 phase (Fig. 5a) with the fluorite-type lattice was 1. Differences between Ba1 – xRx F2 + x Phases obtained from the components by the solid phase reac- with the Distorted Fluorite-type Lattice tion. The Sr2TmF7 phase (an analogue of the Ba2RF7 and Well-Known Ordered Phases phases, Fig. 5b) is an example of ordered phases providing the formation of weak superstructure reflections. Two types of phases were separated in the BaF2–RF3 (R = GdÐLu) systems within the range 0Ð50 mol % RF3 1.3. The phase with a distorted fluorite-type lat- (with the exception of the composition Ba0.625Er0.375F2.375). tice without superstructure reflections on the diffraction pattern was observed and studied on the only 1.1. Cubic nonstoichiometric fluorite-type representative (t'') Ba5Er3F19 = Ba10Er6F38 = Ba1 − xRxF2 + x phases are formed in the systems con- Ba0.625Er0.375F2.375 (different forms of the representation taining all the rare elements listed above (the isother- of its composition in the literature). It has a tetragonally mal cuts were studied in the range from 814 to ° distorted fluorite structure [8] studied in [18, 19]; 1067 C). At the peritectic temperatures, the homogene- sp. gr. I4/mmm, a = 4.199(1) Å, c = 5.986(1) Å, z = 4 ity regions of the Ba1 – xRxF2 + x phases lie within 42Ð (for the composition Ba0.625Er0.375F2.375). The symmetry 31 mol % GdF3 and LuF3, respectively, and are crystal- considerably deviates from cubic; the transition from lized (solid phase synthesis) in the sp. gr. Fm3m with the tetragonal to the pseudocubic unit cell results in the the parameters of the fluorite-type unit cell ranging unit-cell parameter a = 4.1992 = 5.938 Å, which dif- from 5.952 to 6.200 Å. fers by 0.048 Å from the “fluorite” c-parameter.

1.2. Ordered phases with a structure derived A new type of Ba0.75R0.25F2.25 phase with a distorted from the fluorite type are characterized by an fluorite lattice providing no formation of superstructure increased unit-cell volume (in comparison with that of reflections is represented by the Ba0.75Lu0.25F2.25 phase MF2) and superstructure reflections in their diffraction in Fig. 5c. This type differs from the ordered phases dis- patterns. To show that we deal here with a new family cussed in section 1.2 by the absence of weak super- of the Ba0.75R0.25F2.25 phases, we briefly consider the structure reflections on the diffraction patterns. These known ordered phases in the BaF2–RF3 systems. phases differ from Ba5Er3F19 (forming no superstructure reflections either) in composition and metastabil- The (t') Ba RF (R = Dy–Lu, Y) phases with tetrag- 2 7 ity. The Ba5Er3F19 compound, a member of the MnF2n + 6 onal CaF2-type distortion and superstructure reflections homologous series (n = 14), has a definite composition, were obtained by seven-day annealing at 400°C of the whereas Ba0.75R0.25F2.25 is considered only as points in cubic Ba0.67R0.33F2.33 phases synthesized at 1000Ð the series of solid solutions. 1100°C [10, 11]. The above experimental data lead to the following The tetragonally distorted (t''') Ba8 + δR6 – δF34 – δ conclusions: (1) The inhomogeneous chemical compo- phases were discovered in [11] and described for R = sition of as grown Ba0.75R0.25F2.25 rods and the corre- Sm, Gd, and Tb in [8]. These phases are formed from sponding distortions of the cubic fluorite lattice are the trigonal (rhα') phases with an increase in tempera- recorded on the X-ray powder diffraction patterns as ture. Depending on the rare-earth element, their homo- split fluorite reflections. (2) The metastable geneity regions range from 42 to 51 mol % RF3. The Ba0.75R0.25F2.25 (R = GdÐLu) phases with the distorted symmetry of these phases considerably deviates from fluorite lattice have no analogues among the known

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 208 SOBOLEV et al.

I, 103 pulse/s and type of lattice distortion require a further more detailed study. (111) (220) (a) 15 Indices in the fluorite-type cell (CuKα1) 2. Macroinhomogeneity of As Grown Ba0.75 Lu0.25F2.25 (311) synthesized from the components Ba1 – xRxF2 + x Phases by 14-day-annealing at 905°C 10 We believe that distortions of the fluorite lattice are associated with the chemical inhomogeneity of M1 – xRxF2 + x phases resulting from the incongruent melt- (200) ing of the phases of this composition. The M R F (333) 1 – x x 2 + x 5 (400) (422) crystals exhibit two types of macroscopic inhomogene- (331)(420) (531) ity in the distribution of the RF3 component—axial and (442) (222) (440) (620) radial (cellular substructure) distributions. The contribution of axial heterogeneity can be compensated by selecting appropriate samples with a small dimension (111) (220) (b) × 2 along the growth axis. Indices in the fluorite-type cell (CuKα ) 1 As was shown above, the identification of the X-ray 22.5 Sr0.7Tm0.3F2.3 ° diffraction patterns is hampered by their complex form 14-day-annealing at 900 C and the phase composition. The heterogeneity and the tetragonal Sr2TmF7 phase tsub: a = 4.0217 Å; c = 5.7245 Å; anisotropy inherent in some parts of the Ba0.75R0.25F2.25 15.0 tsuper: a = 8.9954 Å; c = 17.175 Å rod were confirmed by crystallooptical studies. The micrographs of the cellular substructure of sam- (311) (200) × 2 ples presented in Fig. 1 are shown in Fig. 6. The cell 7.5 × (400) 2 × shape is better seen on the shadow pattern obtained in 2 (331) × 2 the transmitted light. The refractive-index gradients (222) range within 10−4–10–2 mm–1.

(111) In crossed polarizers, the same parts of the rods (c) show the regions with weak birefringence (2.5 × 10–5– Indices in the ﬂuorite-type cell (CuKα ) −6 1 5 × 10 ), whereas some other parts of the rod are iso- 9 Ba0.75 Lu0.249 Er0.001F2.25 tropic. The observations made at another rotation angle grown from melt of the microscope stage showed that the largest part of the rod is anisotropic (to a different extent) while the

(220) fraction of isotropic parts is small. The average linear 6 (222) dimensions of inhomogeneities vary from 0.2 to 0.7 mm (311) (331) in different samples. (200) (620) These observations are quite consistent with the 3 (333) X-ray diffraction data for the phases with low-symme- (531) (422) (400) (420) (422) try distortions of the fluorite lattice in mixtures contain- (110)? (440) ing cubic phases. The morphology of the cells is different, which is quite understandable because various 20 40 60 80 100 Ba R F phases have different coefficients of θ 0.75 0.25 2.25 2 , deg impurity (R3+) distribution. The honeycomb-like cellular substructure dominates, but, in some instances, the Fig. 5. X-ray diffraction patterns from three phases with a elongated columnar substructure with the cell axes nor- fluorite-type structure in the BaF2–RF3 systems: (a) cubic mal to the growth axis is also observed. The morpho- Ba0.75Lu0.25F2.25 solid solution, (b) ordered tetragonally logical analysis is an individual problem that is distorted Sr2TmF7 phase with a superstructure (an analogue of Ba2TmF7), and (c) a new type of phase with a distorted beyond the scope of the present paper. The mechanism fluorite-type lattice having no superstructure. of formation of a cellular substructure in M1 – xRxF2 + x phases and the methods used to stabilize the planar growth front were discussed elsewhere [20Ð22]. equilibrium phases formed in BaF2–RF3 systems. Systematic morphological studies were made only for (3) New phases are formed in the process of nonequi- the Sr1 – xRxF2 + x family. The morphology of the librium crystallization of incongruent melts. During Ba1 − xRxF2 + x and Ca1 – xRxF2 + x families has not been annealing, these phases often acquire the equilibrium studied as yet. state with a cubic fluorite lattice (at rather high temper- The cell boundaries on the shadow patterns are due atures). We failed to observe the complete transition to light focusing and are not real boundaries between into the equilibrium state. (4) The phase composition the volume parts (grains). At the same time, the shape

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 Ba1 Ð xRxF2 + x PHASES 209

(a)

(b)

3 mm

(c)

Fig. 6. Cellular substructure (radial inhomogeneity of the composition) in the crystal rods in the transmitted (left column) and in the polarized (between crossed polarizers) light: (a) Ba0.75Tb0.249Er0.001F2.25, (b) Ba0.75Tm0.249Er0.001F2.25, and (c) Ba0.75Yb0.249Er0.001F2.25. of these boundaries reﬂects the regular character of the of the crystal lattice). On the whole, the rod remains change of the refractive index in the rod bulk. Far from monolithic (Fig. 1). the point of solid-solution saturation, the boundary is the same solid solution as in the crystal bulk. Between There is no direct relation between the shadow pat- different parts of the rod, the gradients of the refractive tern and the image of the same area obtained in crossed index and other parameters are observed, but no phase polarizers, because these images are formed due to boundary is formed. Macroscopically, the rod consists variations of different quantities (refractive index and of only one phase but is not equilibrium. It consists of birefringence). However, the similar character of mor- mutually penetrating crystallites of different chemical phology (cellular or striated structure) is identical for composition (in the general case, coherent at the level both images, which is especially clear for striated struc-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 210 SOBOLEV et al. tures in Figs. 6a and 6c. The relation between the cellu- determined ambiguously and should be studied in more lar substructure and the distortions of the crystal struc- detail to construct a model consistent with all the exper- ture should be studied separately. imental data. The concentration gradients of R3+ in the cellular Ba R F crystals were not studied. They can be 1 – x x 2 + x 3. Microinhomogeneity of Ba1 – xRxF2 + x Phases estimated indirectly from the X-ray diffraction pattern from the mixture of the pieces of different parts of the The conclusion on the heterogeneity of Ba1 − xRxF2 + x and other M1 – xRxF2 + x phases at the Ba1 – xYbxF2 + x rod. Assuming that the mixture consists of two cubic “phases” (a rather rough approximation, as microlevel was made based on the data on their is seen from Fig. 6c and other figures), the lattice “defect” structure. In these phases, the clusters of defects rather than M2+ and R3+ ions are isomorphously parameters show that there are volumes with a YbF3 concentration of 13.7 and 16.5 mol %. We believe that replaced. These clusters include all types of defects: this difference approximately corresponds to the maxi- highly charged R3+ ions, interstitial F1– ions, and mum difference in the concentrations of the impurity anionic vacancies. The high concentration of R3+ ions in component in various parts of the rod. clusters allows the crystal matrix to preserve the structure and composition close to those of MF . At a Since the cellular substructure (a typical morpho- 2 logical feature of the products of crystallization of the microlevel, the M1 – xRxF2 + x crystals have two chemically different components—an R3+-enriched and Ba1 – xRxF2 + x melts) manifests itself in the distortions of the cubic lattice, it is not improbable that this phenom- R3+-depleted. enon is characteristic of the M1 – xRxF2 + x phases crystal- Hereafter, we propose to call the type of isomor- lized in MF2–RF3 systems (M = Ca, Sr, Ba, Cd, and Pb phism with space filling [3] “block isomorphism.” This and R = Sc, Y, La, and Lu). For Ba1 – xRxF2 + x phases, isomorphism concerns large blocks (hundreds and this phenomenon is the most pronounced because of the thousands of Å3-large volumes) with a close geometry maximum difference in the Ba2+ and R3+ sizes and pro- and crystal structure but possessing different chemical nounced fluctuations of the lattice parameters. compositions instead of isomorphism of ions. The discussion of block isomorphism in M1 − xRxF2 + x phases is As grown Ba1 – xRxF2 + x rods are of a dual nature. beyond the scope of this paper. These are monolithic formations showing no signs of the second phase or the phase boundaries. However, the Recently, heterogeneity at a level of inclusions with X-ray diffraction patterns and the known crystalloopti- a size up to ten parameters of the fluorite unit cell was cal data allow one to consider these rods as a mixture of observed in Ba1 – xLaxF2 + x by the method of diffuse phases with different parameters of undistorted or dis- neutron scattering [23, 24]. These large inclusions of torted fluorite lattices. However, this interpretation is cluster associates can play the role of nuclei of a new inconsistent with an incomplete set of fluorite reflec- phase. However, these crystals also exhibit no phase tions for one of the phases, whereas the mixture of two boundaries and light scattering from the second isostructural and compositionally close phases should phase. have a complete set of reflections for each of the phases. Even in the case of elementary [R6F36] clusters and It might be associated with a relatively small fraction of {M8[R6F68]} superclusters, the blocks that can be iso- the cubic phase (according to the above crystallooptical morphously replaced have linear dimensions of the data). order of nanometers, while their aggregates have dimen- The X-ray diffraction pattern of a Ba0.75Lu0.25F2.25 sions of tens of nanometers. Therefore, M1 – xRxF2 + x powder is fairly well indexed on the assumption of phases can be classified as nanostructurized materials. monoclinic distortion of the fcc lattice (except for sev- The coherence of the nanophase and the matrix (in the eral weak reflections). Only the existence of these addi- general case, the crystal planes of the nanophase are tional reflections does not allow one to index these dif- continuously transformed into the matrix planes) sepa- fraction data as a single phase. There is more than one rate the M1 – xRxF2 + x crystals into an individual class of phase in this powder, which is also confirmed by nanophase materials having no analogues among the images obtained in polarized light. known nanostructurized materials [25]. The justifica- The observations in polarized light (Fig. 6) lead to tion of this classification and its consequences require the model of a nonequilibrium multiphase mixture of special discussion. coherent phases with an undistorted fluorite lattice and Since the nanophase and MF2 matrix are coherent, a distorted optically anisotropic interlayer, which the chemically microheterogeneous M1 – xRxF2 + x crys- removes the stresses caused by the lattice mismatch in tals behave as single crystals irrespective of neutron, the cubic regions in the rod. electron, or X-ray radiation.

Thus, at the present level of our understanding of Electron diffraction from as grown Ba0.8Lu0.2F2.2 Ba0.75R0.25F2.25 phases, their phase composition and the [26] crystals provides different patterns for different character of distortion of their ﬂuorite-type lattice are grains of the rod (Fig. 7). These differences have noth-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 Ba1 Ð xRxF2 + x PHASES 211

[110]c

111c

100 Å [110]c

111F

[213]c

100 Å

Ba0.8Lu0.2F2.2 Fig. 7. Image of the crystal lattice (top) and the corresponding electron diffraction pattern (bottom) of the crystal rod of the nominal (with respect to charge) composition Ba0.8Lu0.2F2.2. ing to do with the decomposition of the sample under stitutional solid solutions, one variety of which is an electron beam. M1 – xRxF2 + x phases. The moiré patterns observed in high-resolution electron micrographs (Fig. 7) are characteristic not only of CONCLUSION BaF2-based Ba1 – xRxF2 + x phases but also of nonstoichiometric fluorites. The moiré fringes can be formed The inhomogeneities in the chemical composition because of the presence of two (or more) phases with of nonequilibrium products of crystallization of incon- different lattice parameters. gruent Ba0.75R0.25F2.25 melts (R = GdÐLu) at the micro- and macrolevels are revealed from their structural and At the coherent conjugation of the cubic fluorite- crystallooptical characteristics. The distortions of cubic type phases with different lattice parameters, elastic fluorite-type lattice in individual Ba R F meta- stresses contribute to lattice distortion and ordering of 0.75 0.25 2.25 the clusters of defects. For solid solutions of metals, the stable phases will be the subject of further detailed theoretical concepts of the decisive role of the deforma- studies. tional interactions in the formation of a microheterogeneous system of periodically alternating regions ACKNOWLEDGMENTS (domains) have been quite well developed ([27], etc.). The deformation-type interactions of the replacing The authors are grateful to L.S. Garashina and atoms were shown to be especially important in sub- V.A. Stasyuk for their help in preparing some experi-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 212 SOBOLEV et al. mental samples and to V.N. Molchanov and 13. B. P. Sobolev, L. S. Garashina, and V. B. Alexandrov, in L.P. Otroshchenko for useful discussions. Proceedings of the 3rd International Congress on Crys- The work was supported by the International Asso- tal Growth, Marseilles, 1971, p. 168. ciation for Promotion of Cooperation of the Former 14. N. L. Tkachenko, L. S. Garashina, O. E. Izotova, et al., Soviet Union (INTAS), project no. 97-32045, and J. Solid State Chem. 8 (3), 213 (1973). the Russian Foundation for Basic Research, project 15. M. Kieser and O. Greis, Z. Anorg. Allg. Chem. 469, 164 no. 99-02-18067. (1980). 16. O. Greis and M. Kieser, J. Less-Common Met. 75 (1), 119 (1980). REFERENCES 17. B. A. Maksimov, Kh. Solans, A. P. Dudka, et al., Kristal- 1. B. P. Sobolev, Butll. Soc. Catalana Cienc. Fis., Quim. lografiya 41, 51 (1996) [Crystallogr. Rep. 41, 50 (1996)]. Mat. 12 (2), 275 (1991). 18. A. M. Golubev and B. P. Sobolev, in Proceedings of the 2. B. P. Sobolev, The Rare Earth Trifluorides, Part 1: High- 7th All-Union Symposium on Chemistry of Inorganic Temperature Chemistry of Rare Earth Trifluorides (Insti- Fluorides, Leninabad, 1984 (Nauka, Moscow, 1984), tut d’Estudis Catalans, Barcelona, 2000). p. 112. 3. V. M. Goldschmidt, T. Barth, G. Lunde, and W. Zachar- 19. A. M. Golubev, B. P. Sobolev, and V. I. Simonov, Kristal- iasen, Geochemische Verteilungsgesetze der Elemente lografiya 30, 314 (1985) [Sov. Phys. Crystallogr. 30, 181 VII, in Skrifter udgitt av det Norske Videnskaps-Akademi (1985)]. i Oslo, I, Mathematik-Naturvidenskapelig Klasse (Oslo, 1926), Vol. 1, no. 2, p. 1. 20. T. M. Turkina, P. P. Fedorov, and B. P. Sobolev, Kristal- lografiya 31, 146 (1986) [Sov. Phys. Crystallogr. 31, 83 4. V. B. Aleksandrov and L. S. Garashina, Dokl. Akad. (1986)]. Nauk SSSR 189 (2), 307 (1969) [Sov. Phys. Dokl. 14, 1040 (1970)]. 21. P. P. Fedorov, T. M. Turkina, V. A. Meleshina, and 5. A. K. Cheetham, B. E. F. Fender, and D. Steele, Solid B. P. Sobolev, in Growth of Crystals (Consultants State Commun. 8 (3), 171 (1970). Bureau, New York, 1988), Vol. 17, p. 165. 6. B. A. Maksimov, Yu. B. Gubina, E. L. Belokoneva, et al., 22. P. P. Fedorov, T. M. Turkina, and B. P. Sobolev, Butll. in Proceedings of the 1st National Conference on Crys- Soc. Cat. Cien. 13 (1), 259 (1992). tal Chemistry, Chernogolovka, 1998, Part I, p. 133. 23. F. Kadlec, F. Moussa, P. Simon, and B. P. Sobolev, Solid 7. G. G. Glavin and Yu. A. Karpov, Zavod. Lab. 30 (3), 306 State Ionics 119 (1Ð4), 131 (1999). (1964). 24. F. Kadlec, F. Moussa, P. Simon, and B. P. Sobolev, 8. B. P. Sobolev and N. L. Tkachenko, J. Less-Common Mater. Sci. Eng. 57 (3), 234 (1999). Met. 85 (2), 155 (1982). 25. H. Gleiter, Acta Mater. 48, 1 (2000). 9. O. Greis, Inaugural-Dissertation Thesis (Univ. Freiburg, 26. R. Múñoz, R. Rojas, B. P. Sobolev, and P. Herrero, in West Germany, 1976). Proceeding 29th Reunión Bienal Society Española de 10. O. Greis and M. Kieser, Rev. Chim. Miner. 16 (6), 520 Microscopia Electrónica, Murcia, 1999, p. 317. (1979). 27. A. G. Khachaturyan, The Theory of Phase Transforma- 11. M. Kieser and O. Greis, J. Less-Common Met. 71 (1), 63 tions and the Structure of Solids Solutions (Nauka, Mos- (1980). cow, 1974), p. 384. 12. B. P. Sobolev, N. L. Tkachenko, V. S. Sidorov, et al., in Proceedings of the 5th All-Union Conference on Crystal Growth, Tbilisi, 1977, Vol. 2, p. 25. Translated by A. Zolot’ko

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 213Ð216. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 249Ð252. Original Russian Text Copyright © 2002 by Chernaya, Volk, Verin, Ivleva, Simonov.

STRUCTURE OF INORGANIC COMPOUNDS

Atomic Structure of (Sr0.50Ba0.50)Nb2O6 Single Crystals in the Series of (SrxBa1 Ð x)Nb2O6 Compounds T. S. Chernaya*, T. R. Volk*, I. A. Verin*, L. I. Ivleva**, and V. I. Simonov* * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia e-mail: [email protected] ** Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 117942 Russia Received May 23, 2001

Abstract—Czochralski grown (Sr0.50Ba0.50)Nb2O6 single crystals have been studied by the method of precision X-ray diffraction analysis. The structural characteristics of (SrxBa1 – x)Nb2O6 compounds with x = 0.33, 0.50, 0.61, and 0.75 were analyzed. The distributions of the Sr and Ba atoms over the crystallographic positions are considered depending on their concentration. The establishment of the mechanisms of isomorphous replacements in these solid solutions allows the variation, within certain limits, of crystal properties by changing the Ba/Sr ratio. © 2002 MAIK “Nauka/Interperiodica”.

The solid solutions of the composition (SrxBa1 – x)Nb2O6 analysis of the collected X-ray data set confirmed the have attracted the continuing interest of researchers tetragonal symmetry of the crystal. The unit-cell because of their electro-optical and nonlinear-optical parameters are a = 12.461(2) Å, c = 3.9475(3) Å. The properties [1Ð4]. Precision X-ray diffraction studies of X-ray diffraction symmetry, 4/mmmP–/–b–, corre- these single crystals provide information on mecha- sponds to three space groups—P4/mbm, P4b2 , and nisms of isomorphous replacements on the atomic scale P4bm. Using the (Sr,Ba)Nb O structures studied ear- in these compounds, whereas simultaneous investiga- 2 6 tions of their properties enable one to establish a corre- lier as an analogy and taking into account their physical lation between their structures and properties, which is properties, we examined and confirmed the acentric important for the transition from the phenomenological space group, P4bm. Averaging symmetrically equiva- description of their crystal properties to the correspond- lent reflections, we arrived at 2035 independent struc- ing microscopic theory. These data are also very impor- ture factors. The R-factor of averaging was 2.4%. tant for materials science because they provide targeted All the computations were carried out by the syntheses of these crystals and the modification of PROMETHEUS program package [8]. The structure already existing crystals instead of the use of the trial- model was refined by the full-matrix least-squares and-error approach. method. The difficulties associated with pronounced correlations between the structural parameters (occu- The X-ray diffraction study of (Sr0.50Ba0.50)Nb2O6 pancies of the positions, thermal parameters) were single crystals completes the systematic investigation overcome by step-by-step scanning. The final R-factor of the structures of (SrxBa1 – x)Nb2O6 compounds with x was 3.0% and Rw = 2.6%. The structure parameters are ranging from 0.25 to 0.75. The results of our previous given in the table. studies of (Sr Ba )Nb O [5], [Sr Ba ]Nb O 0.33 0.67 2 6 0.61 0.39 2 6 The atomic structure of the title compound shown in [6], and (Sr0.75Ba0.25)Nb2O6 [7] solid solutions and the results obtained in the present investigation allow a reli- Fig. 1 consists of sharing oxygen vertices [NbO6]-octa- able comparative analysis of the characteristic features hedra forming a three-dimensional framework. This of the atomic structures of these compounds and their framework has three types of channels along the four- physical properties. fold axis. The narrowest channels are empty. The medium-sized channels are filled with Sr atoms with a The (Sr0.50Ba0.50)Nb2O6 single crystals were grown probability of 72.0%. The broad channels are statisti- by the Czochralski method. The X-ray diffraction data cally occupied by Ba and Sr atoms. In the were collected on a CAD-4F Enraf-Nonius diffracto- (Ba0.27Sr0.75Nb2O5.78) [9] crystal structure, the Ba and meter (åÓKα radiation, ω scan, within the complete Sr atoms occupy one position with a probability of sphere of the reciprocal space, sinθ/λ ≤ 1.2 Å–1) from a 85%. In the crystal structures studied in this paper, the spherical sample 0.30 mm in diameter. A total of Ba and Sr atomic positions are split, and the distribu- 17 300 reflections with I ≥ 3σ(I) were measured. The tion of strontium atoms correlates with the characteris-

214 CHERNAYA et al.

Coordinates and thermal parameters of the basis atoms in the (Sr0.50Ba0.50)Nb2O6 structure Position Symmetry of Atom Occupancy, % x/ay/bz/cB, Å2 multiplicity the position eq Nb(1) 2 100 mm 0.0 0.5 0.0 0.63 Nb(2) 8 100 1 0.0743(1) 0.2113(1) Ð0.0141(7) 0.75 Sr(1) 2 72.0(3) 4 0.0 0.0 0.4791(8) 0.69 Sr(2) 8 13.1(3) 1 0.1509(5) 0.6852(9) 0.5044(19) 1.79 Ba 4 62.3(3) m 0.1727(1) 0.6727(1) 0.4745(7) 1.39 O(1) 8 100 1 0.3436(2) 0.0061(2) Ð0.0441(24) 2.52 O(2) 8 100 1 0.1391(3) 0.0689(2) Ð0.0497(22) 2.40 O(3) 4 100 m 0.2825(2) 0.7825(2) Ð0.0452(22) 2.41 O(4) 4 50 m 0.0114(8) 0.5114(8) 0.4566(24) 3.37 O(5a) 8 50 1 0.3058(5) 0.4054(6) 0.4541(22) 1.32 O(5b) 8 50 1 0.2859(6) 0.4438(7) 0.4512(18) 1.73

tics of ferroelectric phase transitions observed in these units of the composition (SrxBa1 – x)Nb2O6. For the crystals. The splitting of the statistically occupied bar- compounds with x = 0.33, 0.50, 0.61, or 0.75 studied in ium and strontium positions is illustrated by Fig. 1. In this work, the occupancies of the medium-sized channels the (Sr0.33Ba0.67)Nb2O6 crystal [5] with the lowest stron- by strontium atoms vary only slightly (70.5, 72.0, 72.5, tium concentration, all Sr atoms are located in the and 71.5%, respectively). The barium atoms in the medium-sized channels, whereas the broad channels broad channels are always located in positions with are occupied by barium atoms alone. When writing the multiplicities of four in mirror planes m. If the broad chemical formulas with due regard for the occupancies channels are statistically occupied by both barium and of the crystallographic positions, one has to take into strontium atoms, the Sr atoms are displaced from the account also the position multiplicities indicated in the symmetry planes m toward the general positions with table and the fact that the unit cell contains ﬁve formula the multiplicity 8 [6], and the SrÐBa distances are 0.335, 0.305, and 0.262 Å in the compounds with x = 0.50, 0.61, and 0.75, respectively. Taking into account the position multiplicities, the occupancies of the broad channels by barium atoms were determined as 84.0, 62.3, 48.7, and 30.9%, and the occupancies by strontium atoms, as 0.0, 26.2, 40.4, and 57.2% for the compounds with x = 0.33, 0.50, 0.61, and 0.75, respectively, with the total occupancy being almost unchanged (84.0, 88.5, 89.1, and 88.1%, respectively). At x < 0.25 or x > 0.75, crystallization yields phases with structures different from those considered above. The phase we are interested in requires the ﬁlling of the medium-sized channels (the position on the fourfold symmetry axis with the multiplicity 2) with strontium atoms for 71Ð 72%. This occupancy is provided by the presence of at least 25 at. % of strontium in the compound. At higher concentrations of strontium in solid solutions, strontium replaces barium in the broad channels (as was mentioned above, strontium atoms are displaced from y the plane m), whereas its amount in the medium-sized channels remains virtually unchanged. The total occupancy of the broad channels remains almost constant x (~88%). The optically perfect crystals of these com- Sr Ba pounds were obtained from the melt of the congruent composition with x = 0.61. In terms of the structure, the latter crystals are characterized by the occupancy of the Fig. 1. The (Sr0.50Ba0.50)Nb2O6 structure projected onto the ab plane. broad channels by Ba and Sr atoms in a ratio close

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 ATOMIC STRUCTURE 215

Fig. 2. The (Sr0.50Ba0.50)Nb2O6 structure projected onto the ac plane. For notation, see Fig. 1. to 1 : 1. Our results show that the occupancies of broad located in the trans position with respect to the strong channels by Ba and Sr atoms are 48.7 and 40.4%, bond. It is these pairs of the shortest and the longest respectively. bonds that are approximately parallel to the c-axis. In It is seen from Fig. 2 that Sr and Ba ions are located the (Sr0.50Ba0.50)Nb2O6 structure, the Nb(1)ÐO bond at the same height along the c-axis, whereas Nb-octahe- lengths are 1.814 (short bonds), 1.958 (four intermedi- dra are above and below the planes of their location at ate bonds), and 2.155 Å (long bond). A Nb(2)-octahe- distances of c/2. The characteristic feature of Nb-octa- dron has short (1.878 and 1.852 Å bonds with each one hedra is the location of oxygen atoms in the fixed equa- occurring with a probability of 0.5), four intermediate torial positions in all the structures studied, whereas the (1.932, 1.955, 2.000, and 2.000 Å) bonds, and elon- apical oxygen atoms located in the planes of Sr and Ba gated (2.123 and 2.125 Å) bonds. In the compounds atoms are statistically disordered over two positions with x = 0.33, 0.50, 0.61, and 0.75, the average values with a probability of 0.5. In the compound with x = of the short NbÐO bonds are 1.845, 1.840, 1.863, and 0.50, these positions are spaced by the distances of 1.933 Å, respectively; the average values of the inter- 0.407 and 0.521 Å for Nb(1) and Nb(2), respectively. mediate bonds range within 1.961–1.970 Å; and the The distances between the split positions increase elongated NbÐO bond are 2.158, 2.135, 2.108, and monotonically with an increase in the strontium con- 2.048 Å, respectively. Earlier, it was shown [7] that the tent. Thus, these distances are 0.388 and 0.510 Å in the refractive index and some other optical characteristics compound with x = 0.33, whereas the corresponding correlate with the asymmetry of Nb-octahedra distances in the compound with x = 0.75 are 0.561 and expressed as the difference between the short and the 0.635 Å. In other words, the planes statistically occupied by Sr and Ba atoms also contain statistically long NbÐO bonds. The degree of the distortion of arranged apical oxygen atoms of the Nb-octahedra. [NbO6]-octahedra from an ideal centrosymmetric octahedron depends on the isomorphous replacements. The structure of [NbO6]-octahedra is essential for With a decrease in the Sr content, the structure is the optical properties of these crystals. In all structures ordered: the Ba and Sr positions are split, and the of the series under consideration, a Nb(1)-octahedron NbO -octahedra forming the framework become more has the crystallographic symmetry mm, whereas a 6 Nb(2)-octahedron is in the general position. All Nb- distorted (acentric). It can be assumed that this struc- octahedra are aligned along the c-axis and form ÐO(4)Ð tural ordering is responsible for both the dependence of Nb(1)ÐO(4)ÐNb(1)Ð and ÐO(5)ÐNb(2)ÐO(6)ÐNb(2)Ð the relaxation properties on the composition and the chains. As was mentioned above, the bridging oxygen less diffuse phase transition. The filling of the tetrago- atoms of the chains statistically occupy two positions. nal channels with strontium atoms is virtually indepen- As a result, the octahedra “oscillate” about the c-axis of dent of the composition, which governs only the statis- the crystal. The structure of Nb-octahedra can be tical filling of the channels with barium and strontium described by scheme 1Ð4Ð1 showing the existence of atoms [7]. Thus, the properties of (Sr,Ba)Nb2O6 crys- one short (strong) NbÐO bond, four intermediate NbÐO tals can be changed by varying the Sr/Ba ratio in these bonds, and one elongated NbÐO bond, always being structures.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 216 CHERNAYA et al.

ACKNOWLEDGMENTS 4. M. P. Trubelja, E. Ryba, and D. K. Smith, J. Mater. Sci. 31, 1435 (1996). This study was supported by the Federal Program on 5. A. E. Andreœchuk, L. M. Dorozhkin, Yu. S. Kuz’minov, Support of Prominent Scientists and Leading Scientific et al., Kristallografiya 29, 1094 (1984) [Sov. Phys. Crys- Schools, project no. 00-15-96633 and by the Russian tallogr. 29, 641 (1984)]. Foundation for Basic Research, project no. 00-02- 6. T. S. Chernaya, B. A. Maksimov, I. A. Verin, et al., Kri- 16624. stallografiya 42, 421 (1997) [Crystallogr. Rep. 42, 375 (1997)]. 7. T. S. Chernaya, B. A. Maksimov, T. R. Volk, et al., Fiz. REFERENCES Tverd. Tela (St. Petersburg) 42, 1668 (2000) [Phys. Solid 1. L. I. Ivleva and Yu. S. Kuz’minov, Preprint No. 93, FIAN State 42, 1716 (2000)]. (Lebedev Institute of Physics, Academy of Sciences of 8. U. Zuker, K. Perenthalter, W. F. Kuhs, et al., J. Appl. USSR, Moscow, 1977), p. 14. Crystallogr. 16, 358 (1983). 2. M. E. Lines and A. M. Glass, Principles and Applica- 9. P. B. Jamison, S. C. Abrahams, and J. L. Bernstein, tions of Ferroelectrics and Related Materials (Oxford J. Chem. Phys. 42, 5048 (1968). Univ. Press, Oxford, 1977; Mir, Moscow, 1981). 3. L. E. Cross, Ferroelectrics 76, 241 (1987). Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 217Ð222. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 253Ð258. Original Russian Text Copyright © 2002 by Belokoneva, Al’-Ama, Dimitrova, Kurazhkovskaya, Stefanovich.

STRUCTURE OF INORGANIC COMPOUNDS

Synthesis and Crystal Structure of New Carbonate NaPb2(CO3)2(OH) E. L. Belokoneva, A. G. Al’-Ama, O. V. Dimitrova, V. S. Kurazhkovskaya, and S. Yu. Stefanovich Faculty of Geology, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: [email protected] Received June 7, 2001

Abstract—Crystals of a new lead carbonate, NaPb2(CO3)2(OH), sp. gr. P31c, were prepared by hydrothermal synthesis. The crystal structure was established by the heavy-atom method without knowing the exact chemical formula of the compound. The polar structure of the carbonate and the distortion of the pseudosymmetry described by the supergroup P31c are caused by the acentric arrangement of the oxygen atoms providing the satisfactory coordination of Pb and Na atoms. The bonds between a hydroxyl group and two crystallographically independent Pb atoms are directed along the c-axis and have different lengths. The study of the carbonate by the second harmonic generation method in a temperature range of 20Ð250°C revealed the nonlinear optical properties comparable with the similar properties of quartz. The comparison of the structure of the new carbonate with a number of carbonates demonstrated that the new compound is structurally similar to ewaldite BaCa(CO3)2 , diorthosilicate NaBa3[Si2O7](OH), and Ba[AlSiO4]2 containing a double siliconÐoxygen layer. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION autoclave filling was chosen so as to maintain a con- Carbonates belong to one of the most important stant pressure. Colorless transparent hexagonal single classes of inorganic compounds containing planar crystals of sizes up to ~1 mm had a habit ranging from tabular to isometric. The crystals had perfect cleavage 2– anions as the major structural unit. Natural and [CO3] along the (001)-plane. synthetic Ca, Mg, Fe, Mn, Na, Ba, Zn, and Cu carbonates are the most widespread and well-known com- No analogues of the X-ray diffraction powder pat- pounds of this type [1]. Lead carbonates are rather rare tern (a DRON-UM1 diffractometer, Co radiation, and are found in nature as minerals such as phosgenite 40 kV, 25 mA) were found in the PDF data base, which Pb2(CO3)Cl2, cerussite PbCO3, and hydrocerussite indicated that we synthesized a new compound. The composition of the crystals was determined by qualita- PbCO3 á PbO(H2O)2. Due to the high electron polarizability of Pb2+ ions explained by their specific elec- tive X-ray spectral analysis on a CAMSCAN 4DV tronic structure, the constituent oxides of lead carbon- scanning electron microscope equipped with a LINK ates have a dipole structure and possess piezo- and attachment for the energy-dispersive analysis at the pyroelectric and nonlinear-optical properties. Hence, Department of Petrography of the Faculty of Geology the preparation and the study of polar compounds in the of Moscow State University. This investigation con- carbonate systems involving lead are of interest in firmed the presence of Na and Pb atoms in the sample. terms of both crystal chemistry and materials science. Thus, with due regard for the composition of the initial system, the new compound could be related either to carbonates or borates. EXPERIMENTAL Synthesis of Single Crystals and Their Preliminary IR Spectroscopy Diagnostics Single crystals were synthesized when studying the The IR spectrum of a liquid sample obtained as a neat thin film between KBr supporting plates was mea- phase formation in the Na2CO3ÐPbOÐB2O3ÐH2O sys- 3 sured on a Specord-75 IR spectrophotometer in the fre- tem. The experiments were carried out in 5 to 6-cm - Ð1 large standard fluoroplastic-lined autoclaves at ~70 atm quency ranges 1800Ð400 and 3800Ð3000 cm (Fig. 1). and 270Ð280°C. The lower temperature limit was dic- The IR spectrum has stretching vibration bands of the Ð1 tated by the kinetics of hydrothermal reactions, carbonate ion. Thus, the intense band at 1432 cm cor- ν whereas the upper one was determined by the apparatus responds to the asymmetric stretching vibration 3, the used. The experiments were carried out for 18Ð20 days weak band at 1055 cmÐ1 belongs to the symmetric ν to bring the reactions to completion. The coefficient of stretching vibration 1, and the bands at 847 and

218 BELOKONEVA et al.

475

3480

1055

847

695

1432

3600 32001600 1400 1200 1000 800 600 400 ν, cm–1

Fig. 1. IR spectrum of NaPb2(CO3)2(OH).

695 cmÐ1 are attributed to the out-of-plane and in-plane Single-Crystal X-ray Diffraction Analysis deformation vibrations ν and ν , respectively. By anal- 2 4 The symmetry of the Laue patterns obtained from a ogy with the spectrum of cerussite PbCO3 [2], the band × × Ð1 large (0.6 0.3 0.15 mm) transparent tabular single at 475 cm was assigned to lattice vibrations. The crystal oriented normally to the incident beam showed vibration band of the hydroxyl group is observed at Ð1 ν ν that the crystal belongs to the diffraction class 31m . 3480 cm . The degeneration of the 3 and 4 vibrations is indicative of the positional symmetry with a threefold The X-ray data were collected from a small isometric (0.125 × 0.125 × 0.15 mm) well-faceted crystal provid- axis for a CO3 ion. The presence of the stretching vibration ν shows that the symmetry group of a CO ion ing the formation of high-quality reflections in the Laue 1 3 diffraction patterns. The parameters of the triclinic unit contains neither the inversion center nor twofold axes perpendicular to the threefold axis. Thus, the carbonate cell were determined and refined on a Syntex P1 dif- ion has the positional symmetry C3. The superposition fractometer (Table 1). The three-dimensional set of of the resonance vibrations of all the CO3 ions in the intensities Ihkl were collected within the independent unit cell does not change the number of the bands in the region of the reciprocal space in accordance with the spectrum. Consequently, the crystal structure may diffraction class. The intensities were processed and belong to the symmetry class C3 or C3v. 2 converted into Fhkl using the PROFIT program [3]. All the subsequent calculations were carried out with the Table 1. Crystallographic data for NaPb2(CO3)2(OH) and use of the CSD program package. The observed sys- details of X-ray diffraction study tematic absences of the hhl reflections with l = 2n indi- 4 cated the space group P31c–C . The interatomic vec- Molecular formula HC2O7NaPb2 3v Sp. gr. P31c tors of the Patterson function Puvw showed that a, Å 5.268(4) the heavy Pb atoms occupy special positions on the threefold axes spaced by the distance of ~1/3c. The c, Å 13.48(1) refinement of two basis Pb(1) and Pb(2) atoms con- V, Å3 324.0(7) verged with the satisfactory R factor. The Na atom, the ρ 3 O(1) atom of the hydroxyl group (involved in the coor- calc, g/cm 5.877(9) µ, cmÐ1 512.1(1) dination environment only about the Pb(1) and Pb(2) atoms), two C atoms, and the O(2) and O(3) atoms Refinement mode F(hkl) forming the triangular coordination environments Weighting scheme w = 1/[(σ(F))2 + 0.0010F2] about the C atoms were localized from the difference Number of parameters in the 21 electron-density syntheses by the method of successive refinement approximations. The resulting formula 2θ (deg) and sinθ/λ 93.55 and 1.025 NaPb2(CO3)2(OH) (I) is electrically neutral. The struc- max ture model was refined with allowance for anomalous Number of reflections with 851 F > 4σ(F) absorption of the Mo radiation by lead atoms using only isotropic thermal parameters. Attempts at aniso- R(F), Rw 0.0445, 0.0472 tropic refinement failed because of the pronounced S 1.080 pseudosymmetry of the structure in the space group Reduction coefficient 0.685(4) P31c , which is seen from the analysis of the atomic

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 SYNTHESIS AND CRYSTAL STRUCTURE 219

(a) (b) c

Na Rb y Na x

Pb(2)

Pb(1) (c)

Rb(2)

(OH) a

Fig. 2. Crystal structure of NaPb2(CO3)2(OH): (a) the side diagonal projection onto the (110) plane (CO3-triangles are shown by solid lines and Na and Pb atoms are represented by circles); (b) the projection onto the (001) plane (CO3 groups are shown by triangles and Na and Pb atoms are represented by prisms and circles, respectively); (c) the projection onto the (001) plane (the Pb(2)- polyhedra are visible; OH groups are represented by circles). coordinates. A singular point on the pseudoaxis 3 (an in the supergroup was too high and (2) the PbÐO inter- inversion pseudocenter) is located at a height z = 0.14 atomic distance (1.908 Å; the ionic radius of Pb2+ is (and 0.64). Two positions of Pb atoms in the sp. 1.20 Å) and the too long Na–O interatomic distance gr. P31c are related by this pseudocenter, which corre- (3.04 Å; the ionic radius of Na+ is 0.97 Å) were too sponds to one independent Pb atom in the position 4f of short. The splitting of the position 12i of the oxygen the supergroup P31c . In this supergroup, a Na atom atom into two positions, O(2) and O(3), substantially (with the z coordinate rounded to 1/4) occupies the 2c improved the Pb–O (2.62–2.74 Å) and Na–O (2.38– position, a Cl atom is located in the 4e position, the 2.59 Å) distances, reduced the difference in the lengths O(1) atom (with the z coordinate also rounded to ~1/4) of the bonds between the Pb(1) and Pb(2) atoms and the occupies the 2d position, and the O(2) and O(3) atoms common O(1) atom of the hydroxyl group, and lowered merging into one atom are located in the general posi- the symmetry (the loss of the inversion center), which tion 12i. The refinement of this model within the sp. gr. is in agreement with the IR spectral data. At the final P31c did not allow us to reduce the R factor below stage of the refinement, the correction for absorption, 0.087. The supergroup was rejected because of the vio- too high despite the small size of the sample, was lation of two principal structural criteria—(1) the iso- applied using the DIFABS program [4]. The atomic tropic thermal parameter (B = 4.42 Å2) for the O atom coordinates and interatomic distances are given in with the coordinates corresponding to the position 12i Tables 2 and 3, respectively.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 220 BELOKONEVA et al. respectively. The low SHG signal and an insignificant c increase in its intensity for the coarse-ground sample indicate that carbonate I exhibits rather weak optical nonlinearity and, probably, possesses no phase syn- chronism with the neodymium laser radiation and its second harmonic. According to Kurts [6], crystals I should be related to the nonlinear-optical class D. A rise in the temperature to 220Ð250°C resulted in no substantial changes in the nonlinear-optical proper- Ba ties. In the temperature range from 250 to 300°C, the intensity of the second-harmonic signal rapidly decreased and finally became zero. Subsequent cooling did not restore the nonlinear-optical activity of the samples, which proves the irreversibility of the structural change Ca occurred. X-ray diffraction analysis and IR spectroscopy data show that, upon thermal treatment, carbonate I lost the quadratic optical nonlinearity which can be associated with the removal of the OH group from the sample. Fig. 3. Crystal structure of ewaldite projected onto the (110) plane; Ca and Ba atoms are represented by circles, CO3-tri- RESULTS AND DISCUSSION angles are shown by solid lines, and unlocalized CO3 groups are indicated by dashed lines. In the structure of new carbonate I (Figs. 2a, 2b), the CO3-triangle has the symmetry 3, which is consistent with the results of IR spectroscopy. The plane of the Study by Method of the Second Harmonic CO -triangle is parallel to the ab-plane of the unit cell. Generation 3 The CÐO distances are slightly different (Table 3) The nonlinear-optical activity of carbonate I was but range within the corresponding values known for a examined using powdered samples obtained by grind- number of minerals. The CÐO distances in dawsonite ing. A sample with 100 to 150-µm particles was pre- NaAl(OH)2CO3 [7] are 1.25 and 1.308 Å (two dis- pared by isolating the coarse-ground fraction with the tances); in fairchildite K2Ca(CO3)2 [8], 1.284 (two dis- use of sieves. A sample of the finely dispersed fraction tances) and 1.288 Å; in paralstonite BaCa(CO3)2 [9], with 3 to 5-µm grains was obtained by 20-min precipi- 1.279, 1.289, and 1.285 Å; and in barytocalcite tation of an alcoholic suspension. The temperature BaCa(CO3)2 [10], they range from 1.15 to 1.43 Å. The dependence of the second harmonic generation (SHG) average CÐO distance for two triangles in carbonate I is was obtained on the samples placed into an electric fur- identical to that in dolomite CaMg(CO3)2 [11] (CÐO = nace. The test for the second harmonic generation was 1.285 Å). The Pb atoms in the minerals phosgenite performed in the reflection mode using a YAG : Nd Pb (CO )Cl [12], cerussite PbCO [13], and hydrocer- laser according to the procedure described in our earlier 2 3 2 3 ussite Pb (CO ) (OH) [14] with known structures studies [5]. At room temperature, the second-harmonic 3 3 2 2 intensity with respect to that of the reference sample of adopt high coordination numbers (9 or 10). If seven O crystalline quartz with a dispersity of 3 µm were 1.2 atoms located at distances up to ~2.7 Å (Table 3) are and 0.7 for the coarse- and finely-dispersed samples, included into the coordination sphere of Pb atoms, the polyhedra remain open, like those in hydrocerussite; however, the addition of three more O atoms located at Table 2. Coordinates of basis atoms and isotropic thermal distances larger than 3 Å complements the coordination parameters for the NaPb2(CO3)2(OH) structure of the Pb(1) and Pb(2) atoms to ten-vertex polyhedra (because of the pseudosymmetry). These polyhedra can Atom x/ay/bz/cBj be described by hexahedral “boxes” with flat bottoms Na 2/3 1/3 0.8927(11) 1.59(6) and caps ended with shared hydroxyl groups closest to the Pb(1) and Pb(2) atoms (Fig. 2c). The distances in Pb(1) 2/3 1/3 0.5570 0.85(4) these polyhedra around lead atoms in phosgenite (from Pb(2) 2/3 1/3 0.2258(1) 0.90(4) 2.36 to 3.34 Å), cerussite (from 2.59 and 2.76 Å), and C(1) 0 0 0.035(3) 0.44(6) hydrocerussite (from 2.75 to 3.10 Å) are quite consistent C(2) 0 0 0.276(3) 0.32(6) with the PbÐO distances in carbonate I refined within the acentric space group. The coordination polyhedra around O(1) 2/3 1/3 0.388(4) 1.47(6) Na atoms are almost regular trigonal prisms. O(2) 0.151(7) 0.288(8) 0.536(2) 1.20(6) The metrics of the axes in the structures of carbonate I O(3) 0.276(7) 0.141(7) 0.755(2) 0.83(6) and ewaldite BaCa(CO3)2 (a = 5.284 Å, c = 12.78 Å,

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 SYNTHESIS AND CRYSTAL STRUCTURE 221 sp. gr. P63mc [15]) are virtually identical. The replace- (a) ment of Ba2+ by Na+ leads to a necessary increase in the number of large cations in the formula of carbonate I. One of the CO3-triangles in the ewaldite structure has not been localized as yet [15]. Comparing the side diagonal projections of these structures (Figs. 2a, 3), we see that the positions of Ba and Ca atoms in ewaldite are identical to the positions of Na and Pb(1) atoms in car- b Ba(2), Na bonate I. By analogy, it can be assumed that a CO3 Ba(2) group not revealed in the ewaldite structure should be located between the Ca- and Ba-layers (dashed lines in Fig. 3) at the height z identical to the height observed for an additional Pb(2) cation in structure I. a The mineral paralstonite of the aragonite group with the formula BaCa(CO3)2 identical to that of ewaldite (b) belongs to the trigonal system (sp. gr. P321) [9]. The c- parameter of paralstonite (6.148 Å) is half as large as that of carbonate I, whereas the a(b) parameter Ba(1) (8.692 Å) is close to the length of the unit-cell diagonal of carbonate I. The close a(b) parameters were also found in phosgenite and hydrocerussite. The cationic a Si b motifs in ewaldite and carbonate I are complicated by Ba(2) an additional layer built into it and increasing in the c- parameter. The unit cell of the minimum volume (sp. gr. P312) is observed in lead- and oxygen-deficient borate Pb0.75[BO2.25] studied recently. The a(b) parameters of the latter (~5 Å) are close to those of aragonite, fairchildite, dolomite, and cerussite, whereas its c parame- Fig. 4. Crystal structure of NaBa3[Si2O7](OH) (a) projected ter is equal to that of ewaldite. This borate was found to onto the (001) plane (Ba(2)-polyhedra are seen; Ba(1) and Na atoms are represented by overlapping circles); and be structurally close to carbonates, and, ﬁrst of all, to (b) projected onto the (001) plane (Ba(1)-prisms and the Si aragonite. diortho groups are seen; Ba(2) atoms are represented by circles). In the search for compounds structurally close to carbonate I, we revealed, quite surprisingly, diorthosilicate NaBa3[Si2O7](OH) [16] (II), whose unit-cell the pyramid occupied by OH groups with shortened parameters (a = 5.79 Å, c = 14.74 Å, sp. gr. P63/mmc) BaÐOH distances (Fig. 4a) (as in the structure of car- are very close to those of I (despite the differences bonate I). In carbonate I, Na-prisms are observed between these compounds). The coordination polyhe- instead of diortho [Si O ] groups in II (which link dron around Ba(2) in II is identical to Pb-polyhedra in 2 7 Ba(2)-polyhedra into columns), with the Na atoms in I (within the symmetry of the position). This polyhedron was described as the combination of a hexagonal the prisms occupying the central position of the bridg- pyramid and a half of the Archimedean cuboctahedron, ing O atom of a diortho group in II. The columns of the whose triangular base is formed by a triangular face of second type in II along the c-axis consist of Na-octahe- the tetrahedron of a linear diortho group. In II, Ba(2)- dra and Ba(1)-trigonal prisms (Fig. 4b). In carbonate I, polyhedra are linked in pairs via the shared vertices of these columns are replaced by CO3-triangles (Fig. 2b)

Table 3. Principal interatomic distances (Å) in the NaPb2(CO3)2(OH) structure Pb(1)-polyhedron Pb(2)-polyhedron Na-polyhedron Pb(1)ÐO(1) 2.28(5) Pb(2)ÐO(1) 2.18(5) NaÐO(3) 2.38(4) × 3 Pb(1)ÐO(2) 2.62(4) × 3 Pb(2)ÐO(3) 2.72(4) × 2 NaÐO(2) 2.59(4) × 3 Pb(1)ÐO(3) 2.70(4) × 3 Pb(2)ÐO(3) 2.74(4) × 3 Pb(1)ÐO(3) 3.44(5) × 3 Pb(2)ÐO(2) 3.09(5) × 3 C(1)-triangle C(2)-triangle C(1)ÐO(2) 1.31(4) × 3 C(2)ÐO(3) 1.26(4) × 3 O(2)ÐO(2) 2.28(5) × 3 O(3)ÐO(3) 2.18(5) × 3

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 222 BELOKONEVA et al. analogy between diorthosilicates and minerals with triangular anions [18].

b' Ba a' ACKNOWLEDGMENTS We are grateful to E.A. Guseva (the Department of a Si b Petrography of the Faculty of Geology of Moscow State University) for performing the X-ray spectral analysis.

REFERENCES 1. I. Kostov, Mineralogy (Oliver and Boyd, Edinburgh, 1968; Mir, Moscow, 1971). Fig. 5. Crystal structure of Ba[AlSiO4]2 projected onto the (001) plane; Si-tetrahedra are seen; Ba atoms are shown by 2. The Infrared Spectra of Minerals, Ed. by V. C. Farmer circles. The a' and b' axes are shown to facilitate the com- (Mineralogical Society, London, 1974), p. 206. parison of the structures. 3. V. A. Strel’tsov and V. E. Zavodnik, Kristallograﬁya 34, 1369 (1989) [Sov. Phys. Crystallogr. 34, 824 (1989)]. (the shared bases of the octahedra and prisms in II). In 4. N. Walker and D. Stuart, Acta Crystallogr., Sect. A: carbonate I, all the distances between the triangles Found. Crystallogr. A39 (1), 158 (1983). along the c-axis are equivalent (Fig. 2a), whereas the 5. S. Yu. Stefanovich, in Extended Abstracts of European corresponding distances in II have different values due Conference on Lasers and Electro-Optics (CLEO- Europe’94), Amsterdam, 1994, p. 249. to the differences in the heights at which the Na-octahedron and the Ba(1)-prism are located. Similar to the 6. S. K. Kurts and T. T. Pery, J. Appl. Phys. 39 (8), 3798 shared bases of the octahedra and prisms in II (Fig. 4b), (1968). 7. E. Corazza, C. Sabelli, and S. Vannucci, Neues Jahrb. the pairs of CO3-triangles in the structure of I (Fig. 4b) are rotated with respect to one another by 60°. The Mineral., Monatsh., 381 (1977). 8. F. Pertlik, Z. Kristallogr. 157, 199 (1981). NaBa3[Si2O7](OH) silicate is structurally similar to the NaPb (CO ) (OH) carbonate because the ratio of the 9. H. Effenberger, Neues Jahrb. Mineral., Monatsh., 353 2 3 2 (1980). ionic radii of the pair Pb2+ and C4+ is similar to the ratio 10. K. F. Alm, Ark. Mineral. Geol. 2, 399 (1960). of the pair Ba2+ and Si4+. The fragments of these structures shown in Figs. 2b and 4b are rather similar within 11. H. Effenberger and K. Mereiter, Z. Kristallogr. 156, 233 the framework of the above-considered topology (Na- (1981). 12. G. Giuseppetii and C. Tadini, TMPM, Tschermaks Min- prismÐSi-tetrahedron, CO3-triangleÐBa(1)-prism, and PbÐBa(1)) and can be approximated by an idealized sil- eral. Petrogr. Mitt. 21, 101 (1974). iconÐoxygen double layer built by the simplest mica 13. G. Chevrier, G. Giester, G. Heger, et al., Z. Kristallogr. [Si O ] layers linked by shared apical vertices. Such a 199, 67 (1992). 4 10 14. J. M. Cowley, Acta Crystallogr. 9, 391 (1956). layer was found in the Ba[Al2Si2O8] structure [17] (Fig. 5). These layers are translated and, thus, the 15. G. Donnay and H. Preston, TMPM, Tschermaks Min- c-parameter in the latter structure (7.79 Å) is half as eral. Petrogr. Mitt. 15, 201 (1971). large as that in the structures under consideration. In 16. O. S. Filipenko, E. A. Pobedimskaya, V. I. Ponomarev, this layer, the diortho groups can be replaced either by and N. V. Belov, Dokl. Akad. Nauk SSSR 200, 591 trigonal prisms (of Ba or Na) or other anionic tetrahe- (1971) [Sov. Phys. Dokl. 16, 703 (1972)]. 17. Y. Takeuchi, Mineral. J. 2, 311 (1958). dral or triangular (CO3) groups, with the latter replacing 18. N. V. Belov, Essays on Structural Mineralogy (Nedra, the triangular bases of the SiO4-tetrahedra. This fact opens new possibilities for crystallochemical analysis Moscow, 1976). and the prediction of new structures. It should be noted that N.V. Belov was the ﬁrst to analyze the structural Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 223Ð228. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 259Ð264. Original Russian Text Copyright © 2002 by Belovitskaya, Pekov, Gobechiya, Yamnova, Kabalov, Chukanov, Schneider.

STRUCTURE OF INORGANIC COMPOUNDS

Crystal Structures of Two Ancylite Modifications Yu. V. Belovitskaya*, I. V. Pekov*, E. R. Gobechiya*, N. A. Yamnova*, Yu. K. Kabalov*, N. V. Chukanov**, and J. Schneider*** * Geology Faculty, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: [email protected] ** Institute of Problems of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia *** Institut für Kristallograﬁe und Mineralogie, University of Münich, Münich, D-80-333 Germany Received July 6, 2000

Abstract—The crystal structures of two ancylite specimens from Khibiny massif (the Kola Peninsula, Rus- sia)—ancylite-(Ce) from alkali hydrothermalites

(Sr1.01Ca0.02Ba0.01)Σ1.04(Ce0.52La0.28Nd0.11Pr0.04Sm0.01)Σ0.96(CO3)2(OH0.83F0.13)Σ0.96 á 0.9H2O and ancylite-(Ce) from carbonatites—have

(Sr0.80Ca0.05Ba0.01)Σ0.86(Ce0.62La0.40Nd0.09Pr0.03)Σ1.14(CO3)2(OH0.99F0.15)Σ1.14 á 1.0H2O been reﬁned by the Rietveld method. A focusing STOE-STADIP diffractometer with a bent Ge(111) primary monochromator was used (λMoKα radiation, 2.16° < 2θ < 54.98°; reﬂection number 237–437). All the computations for 1 ancylite from alkali hydrothermalites were performed within the sp. gr. Pmc21, a = 5.0634(1) Å, b = 8.5898(1) Å, c = 3 7.2781(1) Å, V = 316.55(1) Å , Rwp = 1.90; the computations for ancylite from carbonatites were performed within 3 the sp. gr. Pmcn, a = 5.0577(1) Å, b = 8.5665(2) Å, c = 7.3151(2) Å, V = 316.94(1) Å , Rwp = 2.38 in the anisotropic approximation of thermal vibrations of cations and oxygen atoms. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION Tulilukht Bay) described by the formula Ancylite-(Ce) described by the formula (Sr0.80Ca0.05Ba0.01)Σ0.86(Ce0.62La0.40Nd0.09Pr0.03)Σ1.14 á ≈ (Sr,Ca)2 − xREEx(CO3)2(OH,F)x á nH2O (with x 1.0–1.5, (CO3)2(OH0.99F0.15)Σ1.14 á 1.0H2O (sp. gr. Pmcn) (speci- n ≈ 1) is one of the rare earth carbonates occurring in men 54). Since the IR spectra obtained from these two many alkaline massifs (the Khibiny, Sebljavr, the Kola peninsula, Vuorijarvi, Sallanlatva, and Northern Kare- lia (Russia), Mont Saint-Hilaire (Canada), Narssarssuk (Greenland), etc.). It is also characteristic of late low- alkali hydrothermal formations associated with agpaitic rocks and carbonatites. The ancylite family consists of five minerals described 2+ by the general formula M2 Ð x REEx(CO3)2(OH)x á nH2O, namely, ancylite-(Ce) and ancylite-(La) with M2+ = Sr [1], calcioancylite-(Ce) [2] and calcioancylite-(Nd) [3] with M2+ = Ca, and gysinite-(Nd) [4] with M2+ = Pb. The crystal structure of ancylite-(Ce) was first determined within the sp. gr. Pmcn in 1975 [5]. The metal atoms in the structure are located in ten-vertex polyhedra forming a three-dimensional framework (Fig. 1). We studied crystal structures of two ancylite-(Ce) b modifications found in specimens from Khibiny massif Ò (the Kola Peninsula): the ancylite-(Ce) from alkali hydrothermalites (the Lovchorrite Mine, the Hackman Fig. 1. Projection of an ideal ancylite structure onto the 2Ð Valley) described by the formula (Sr1.01Ca0.02Ba0.01)Σ1.04 á (100) plane. The CO3 groups are indicated by black triangles. The black circles indicate the O(3) position in the (Ce0.52La0.28Nd0.11Pr0.04Sm0.01)Σ0.96(CO3)2(OH0.83F0.13)Σ0.96 á structure of ancylite from carbonatites (specimen 54) corre- 0.9H2O (sp. gr. Pmc21) (specimen 52) and sponding to the O(31) and O(32) positions in the structure ancylite-(Ce) from carbonatites (the region of the of ancylite from alkali hydrothermalites (specimen 52).

1715 after burbankite.

720 The chemical compositions of the specimens (cat- (a) 710 3490 697

1069 ions, ﬂuorine, Table 1) was studied by electron 861 microprobe analysis (Camebax SX 50, analyst N.N. Kononkova, Moscow State University). The powder specimens were prepared from the specially selected homogeneous ancylite grains. The X-ray 1437 1770 1483 738 872 diffraction spectra were obtained on a focusing STOE- 1069 719 STADIP diffractometer with bent Ge(111) primary (b) 712 699

860 λ α θ monochromator MoK 1 radiation, 2.16° < 2 < 3545 1545 54.98°, with a total number of reflections of 437 (spec- 1445 imen 52); 5.00° < 2θ < 44.98°, with a total number of 1520 reﬂections of 237 (specimen 54). All the computations

3500 3000 were made by the WYRIET program, version 3.3 [6] 1422 1482 within the sp. gr. Pmc21 (specimen 52) and the sp. gr. Pmcn (specimen 54). The initial structure model

1500 1000 refined within the sp. gr. Pmcn was the model suggested by Dal Negro, Rossi, and Tazzoli [5]. The model refined within the sp. gr. Pmc21 was also based on the Fig. 2. IR spectra of ancylites from (a) alkali hydrother- structure of highly symmetric ancylite but with due malites and (b) carbonatites. The numbers indicate frequen- regard for the absence of the plane n. cies in cmÐ1. Specimen 54 showed the presence of strontianite impurity (4.2%) and, therefore, the refinement was per- specimens (Fig. 2) were essentially different, we under- formed for two phases simultaneously—ancylite and took their structural investigation. strontianite as the second phase. We used ionic scattering curves. The profile peak was approximated according to the Pearson-VII func- EXPERIMENTAL tion at 6FWHM. The asymmetry was refined at 2θ < 40°. Ancylite-(Ce) from alkali hydrothermalites from the The refinement was performed with gradual inclusion Hackman Valley (specimen 52) forms bright yellow into the process of new parameters to be refined and aggregates of split platelike crystals with a thickness of constant automatic modeling of the background until up to 0.5 mm forming, in turn, cellular pseudomorph the attainment of a stable R factor. Upon the refine- after the hexagonal prismatic mineral (probably bur- ment in the isotropic approximation, the reliability bankite) in association with feldspar, natrolite, eudia- factor Rwp was 2.87% (specimen 52) and 2.67% (spec- lyte, aegirine, astrophyllite, and apatite. Ancylite-(Ce) imen 54). from calcite carbonatites from the region of the Tuli- Some parameters of the data acquisition and the lukht Bay (specimen 54) is an aggregate of white split results of the structure refinement of ancylites in the

Table 1. Chemical composition (wt %) and the formulas of studied ancylites Components Specimen 52 Specimen 54 Components Specimen 52 Specimen 54

CaO 0.34 0.68 Sm2O3 0.31 0.09 SrO 27.65 21.24 F 0.64 0.73

BaO 0.46 0.36 CO2 (23.28) (22.44) La2O3 12.10 16.64 (H2O)1 (1.98) (2.27) Ce2O3 22.50 25.53 (H2O)2 (4.32) (4.92) Pr2O3 1.76 1.37 ÐO=F2 0.27 0.31 Nd2O3 4.93 4.03 Total (100.00) (100.00) Note: 1. Specimen 52 is ancylite from alkali hydrothermalites from Lovchorrite Mine and Hackman Valley of the compositions (Sr1.01Ca0.02Ba0.01)Σ1.04(Ce0.52La0.28Nd0.11Pr0.04Sm0.01)Σ0.96(CO3)2(OH0.83F0.13)Σ0.96 á 0.9H2O. Specimen 54 is ancylite from carbonatites from the region of Tulilukht Bay (Sr0.80Ca0.05Ba0.01)Σ0.86(Ce0.62La0.40Nd0.09Pr0.03)Σ1.14(CO3)2(OH0.99F0.15)Σ1.14 á 1.0H2O. 2. The calculated values are given in parentheses; the number of OH-groups (to which the (H2O)1 value corresponds) was calculated from the stoichiometry; the number of water molecules (H2O)2 was calculated from the difference of the total.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 CRYSTAL STRUCTURES OF TWO ANCYLITE MODIFICATIONS 225

Table 2. Unit-cell parameters and the refined data for crystal structures of two ancylite specimens Characteristic Specimen 52 Specimen 54 Characteristic Specimen 52 Specimen 54 a, Å 5.0634(1) 5.0577(1) Rp 1.45 1.87 b, Å 8.5898(1) 8.5665(2) Rwp 1.90 2.38 c, Å 7.2781(1) 7.3151(2) Rexp 1.83 2.87 3 V, Å 316.55 316.94 RB 1.53 1.34 Sp. gr. Pmc21 Pmcn RF 1.43 1.63 2θ-range, deg 2.16Ð54.98 5.00Ð44.98 s* 1.04 0.83 Number of reflections 437 237 DWD** 0.81 1.04 σ*** Number of refined parameters 90 50x 1.660 1.587

* s = Rwp/Rexp, Rexp is the expected value of Rwp. ** DurbinÐWatson statistics [8]. *** Factor used for calculating standard deviations [9].

Table 3. Atomic coordinates, isotropic temperature parameters (Å2), and occupancies of the positions in ancylite-(Ce) from alkali hydrothermalites (specimen 52), sp. gr. Pmc21 Atom Characteristic Value Atom Characteristic Value

å(1) x 0 O(11) x 0 y 0.0977(5) y 0.418(4) z 0.6437(5) z 0.271(3)

Bj 0.48(5) Bj 2.1(8) q (Sr)* 0.486(2) O(12) x 0.5 q (RE + Ba)** 0.486(2) y 0.919(4) q (Ca) 0.016(6) z 0.244(2)

Bj 1.4(8) M(2) x 0.5 O(21) x 0.277(2) y 0.5897(5) y 0.122(2) z 0.8475(5) z 0.352(2)

Bj 0.66(5) Bj 1.3(5) q (Sr) 0.498(2) O(22) x 0.223(2) q (RE) ** 0.498(2) y 0.617(2) C(1) x 0 z 0.151(2)

y 0.554(6) Bj 1.5(5) z 0.191(2) O(31) x 0

Bj 1.9(9) y 0.204(3) z 0.992(3)

C(2) x 0.5 Bj 1.4(9) y 0.056(6) O(32) x 0.5 z 0.308(2) y 0.660(3)

Bj 1.8(9) z 0.516(3)

Bj 1.0(8)

Notes: 1. Reﬁnement was made for the composition (Sr0.49RE0.48Ca0.02Ba0.01)Σ1.00(Sr0.50RE0.50)Σ1.00(CO3)2(OH, F)0.98 á 0.9H2O. The number of OH-groups was calculated from the stoichiometry; the number of water molecules was calculated from the difference of the total. 2. The reﬁnement for RE was made with the use of the f-curve of cerium.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 226 BELOVITSKAYA et al.

I, 103 pulse/s

(a) 15

10 20 30 40 50

6 (b)

5 10 15 20 25 30 35 40 2θ, deg

Fig. 3. Experimental (solid line) and calculated (dots) X-ray diffraction spectra from ancylites (a) specimen 52 and (b) specimen 54. Arrows indicate the additional reﬂections in the spectrum from ancylite from alkali hydrothermalites (2θ = 4.739°, d = 8.579 Å, hkl = 010 and 2θ = 8.020°, d = 5.072 Å, hkl = 100) which violate the law of systematic absences for the sp. gr. Pmcn. The specimen from carbonatites has a strontialite impurity (the lower line diagram).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 CRYSTAL STRUCTURES OF TWO ANCYLITE MODIFICATIONS 227

Table 4. Atomic coordinates and isotropic temperature parameters (Å2) and occupancies of the positions in ancylite-(Ce) from carbonatites (specimen 54), sp. gr. Pmcn Atom Characteristic Value Atom Characteristic Value Mx 1/4 O(1) x 3/4 y 0.3409(3) y 0.3286(2) z 0.6482(3) z 0.7336(2)

Bj 0.96(2) Bj 2.7(5) q (RE + Ba)* 1.153(3) O(2) x 0.5266(2) q (Sr) 0.798(4) y 0.1223(1) q (Ca) 0.056(8) z 0.8525(2)

Bj 1.2(3) C x 3/4 O(3) x 1/4 y 0.1961(4) y 0.4183(2) z 0.8154(5) z 0.9736(2)

Bj 3.7(9) Bj 2.7(5)

Notes: 1. Refinement was made for the composition (RE1.13Sr0.80Ca0.06Ba0.02)Σ2.01(CO3)2[(OH, F)1.15(H2O)1.0]Σ2.15. The number of OH-groups was calculated from the stoichiometry; the number of water molecules was calculated from the difference of the total. 2. The refinement for RE was made with the use of the f-curve of cerium. anisotropic (metal and oxygen atoms) and isotropic hkl = 010 and 2θ = 8.020°, d = 5.072 Å, hkl = 100), (carbon atoms) approximations are listed in Table 2. which violate the law of systematic absences for the Figure 3 shows the experimental (solid line) and the sp. gr. Pmcn. Therefore, the structure of this mineral calculated (dots) X-ray diffraction spectra of ancylites. was refined within the less symmetric group Pmc21 in The coordinates and the isotropic temperature correc- which the cations (Sr, Ca, Ba, and RE) are located in tions for atoms and the position occupancies are indi- two crystallographically nonequivalent positions M(1) cated in Tables 3 and 4. and M(2). Since the ratio of the number of divalent cat- The IR spectra of ancylites prepared in the shape of ions to the total number of rare earth elements in this tablets with KBr were obtained on a Specord 75 IR specimen was close to 1 : 1, we first assumed that M2+ spectrophotometer. and RE occupy two separate positions and that the lowering of the symmetry is caused by cationic ordering. However, the structure refinement showed that both DISCUSSION these positions are occupied statistically (Table 3) and the lowering of the symmetry in the structure of this The crystal structure of ancylite-(Ce) was first ancylite modification is caused by the displacement of solved within the sp. gr. Pmcn as a derivative of the ara- atoms from the special positions to the general ones. gonite structure in 1975 [5]. Our results obtained for ancylite-(Ce) from carbonatites (specimen 54) agree The disappearance of the plane n is associated with the well with the data of this first structure determination split of each position in the above structure into two [5]. Divalent Sr, Ca, Ba cations and rare earth elements nonequivalent positions (Tables 3 and 4), with the dif- in ancylite are located in ten-vertex polyhedra formed ferences in the coordinates of each pair of atoms being by O(1), O(2), and O(3) oxygen atoms. Ten-vertex much more pronounced than the accuracy of the deter- polyhedra share triangular O(1)ÐO(3)ÐO(2) faces and mination of these coordinates. form chains along the c-axis. In turn, these chains form The calculation of the local balance by the method a three-dimensional framework via (CO3)-triangles, suggested in [7] and the allowance for possible hydro- with each triangle being surrounded with three M-poly- gen bonds allowed us to single out O atoms, (OH)– hedra sharing their edges (Fig. 1). groups, and H2O molecules in the anionic part of the The average distances in the polyhedra are MÐO structure. In the structure of ancylite from carbonatites 2.63, C–O 1.31 Å (specimen 54) and M(1)ÐO 2.69, (specimen 54), the O(1) and O(2) positions are occu- M(2)–O 2.62, C(1)–O 1.29, and C(2)–O 1.29 Å (speci- pied by O atoms, whereas the O(3) position is statisti- – men 52). cally filled with (OH) -groups and ç2é-molecules. In Unlike the X-ray diffraction pattern from ancylite the structure of low-symmetric ancylite from alkali from carbonatites (Fig. 3b), the X-ray diffraction pattern hydrothermalites (specimen 52), the éç–-groups and from ancylite from alkali hydrothermalites (Fig. 3a) has ç2é molecules are orderly distributed over the O(32) some additional reflections (2θ = 4.739°, d = 8.579 Å, and O(31) positions, respectively.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 228 BELOVITSKAYA et al. The IR spectrum of ancylite from alkali hydrother- different formulas (Tables 3 and 4). The representative malites (specimen 52) (Fig. 2a) has a band at 3480Ð sampling of IR spectra of ancylites from numerous 3490 cmÐ1 corresponding to the stretching vibrations of alkaline massifs over the entire world allows us to state OH– groups forming relatively weak hydrogen bonds. that both ancylite modifications occur widely in nature. Sometimes, we also observed a shoulder at 3530Ð Ð1 3540 cm , which is attributed to free OH-groups ACKNOWLEDGMENTS whose number is considerably less than the number of these groups linked by hydrogen bonds. The water mol- This study was supported by the Russian Founda- ecules form rather strong hydrogen bonds, which man- tion for Basic Research, project no. 99-05-39019 and ifest themselves in the IR spectrum as a broad band the National Scientific Foundation of China (NSDC). with the maximum at 2900Ð3000 cmÐ1. The triply 2Ð degenerate band of bending vibrations of CO3 anions REFERENCES is split into four to six components, which indicates the 1. V. N. Yakovenchuk, Yu. P. Men’shikov, Ya. A. Pakho- presence of CO3-groups of at least two kinds. At the movskiœ, and G. Yu. Ivanyuk, Zap. Vseross. Mineral. same time, the presence of an unsplit band due to fully O−va. 126 (1), 96 (1997). symmetric stretching vibrations of CO2Ð groups at 2. I. V. Pekov, O. V. Petersen, and A. V. Voloshin, Neues 3 Jahrb. Mineral., Abh. 171 (3), 309 (1997). 1068 cmÐ1 indicates a considerable violation of the 3. P. Orlandi, M. Pasero, and G. Vezzalini, Eur. J. Mineral. local symmetry of carbonate anions only for one kind 2, 413 (1990). 2Ð of CO3 groups. 4. H. Sarp and J. Bertrand, Am. Mineral. 70 (11Ð12), 1314 The IR spectrum of ancylite from carbonatites (1985). (specimen 54) considerably differs from that of ancylite 5. A. Dal Negro, G. Rossi, and V. Tazzoli, Am. Mineral. 60, from alkali hydrothermalites in that it has a narrow 280 (1975). intense band at 3540Ð3545 cmÐ1 and a narrow band 6. J. Schneider, in Profile Refinement on IBM-PC`s: Pro- at 739 cmÐ1 (Fig. 2b) attributed to the stretching and ceedings of the IUCr. International Workshop on the bending vibrations of OH– anions forming no hydrogen Rietveld Method, Petten, 1989, p. 71. bonds, whereas no hydroxyl groups linked by a hydro- 7. Yu. A. Pyatenko, A. A. Voronkov, and Z. V. Pudovkina, gen bond are present at all (the absence of the band at Mineralogical Crystal Chemistry of Titanium (Nauka, Moscow, 1976), p. 155. 3480Ð3490 cmÐ1). 8. R. Hill and H. Flack, J. Appl. Crystallogr. 20, 356 Thus, we can state that there exist two ancylite mod- (1987). ifications which are crystallized in two different space groups of the orthorhombic system and have essentially 9. J.-F. Berar and P. Lelann, J. Appl. Crystallogr. 24, 1 (1991). different X-ray and IR-spectroscopic characteristics. Taking into account the corresponding crystallochemical data, these modifications should be described by Translated by L. Man

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 229Ð231. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 265Ð266. Original Russian Text Copyright © 2002 by Rozenberg, Rastsvetaeva, Pekov, Chukanov.

STRUCTURE OF INORGANIC COMPOUNDS

Structural Characteristics of a New Cation-Deficient Representative of the Labuntsovite Group K. A. Rozenberg*, R. K. Rastsvetaeva**, I. V. Pekov*, and N. V. Chukanov*** * Faculty of Geology, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia ** Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia e-mail: [email protected] *** Institute of Problems of Chemical Physics in Chernogolovka, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia Received June 6, 2001

Abstract—The crystal structure of a new representative of the labuntsovite group from the Khibiny massif (the Kola Peninsula) has been reﬁned. The unit-cell parameters are a = 14.298(7) Å, b = 13.816(7) Å, c = 7.792(3) Å, β 3 σ = 116.85(5)°, V = 1373.3 Å , sp. gr. C2/m, Raniso = 0.047, 1084 reflections with F > 4 (F). The mineral differs from lemmleinite-Ba and other members of the group by the predominance of the vacancies in two key cationic positions—C position is occupied by Ba cations (37%) and D position is occupied mainly by Mn cations (47%). © 2002 MAIK “Nauka/Interperiodica”.

The minerals of the labuntsovite group are charac- tallochemical formula of the mineral can be written as terized by wide-range compositional heterovalent iso- Na3.5K4[Ba1.5(Mn,Fe)0.9(H2O)1.8][Ti7.8Nb0.2(O,OH)8] á morphism, which should necessarily be taken into [Si4O12]4 á 5.4H2O. account when predicting the ion-exchange properties of The structure of the new mineral (ﬁgure) has a micro- and mesoporous materials based on the minerals mixed framework typical of labuntsovites consisting of of this structure type. Ten representatives of this group have been structurally characterized. In this study, we established the structure of a new specimen from the b Gakman valley (the Khibiny massif, the Kola Penin- sula). The mineral was discovered in pegmatitic bulges of the natroliteÐmicrocline composition embedded in gneissous nepheline syenites in the form of orange short-prismatic crystals up to 5 mm in length. The mineral occurs in association with aegirine, catapleiite, barylite, ﬂuorite, and galenite. The chemical composition of the new mineral was established using seven points by electron microprobe analysis (the water content was not determined) and corresponds to the following empirical formula (calculated for 16 Si atoms):

Na4.09K3.88Ba1.51Mn0.61Fe0.18Ti6.29Nb1.77(O6.58OH1.42) á [Si4O12]4 á 12.75H2O. The principal characteristics of the crystal and the details of the X-ray data collection are given in Table 1. a Proceeding from the chemical composition, we assumed that the new mineral is structurally similar to lemmleinite-Ba and used the coordinates of the framework atoms of the latter [1] as the starting model. The KNaH2O Ba, H2O difference electron density synthesis calculated upon the reﬁnement of the structure revealed the split K(1) Crystal structure of the mineral projected onto the (001) and K(2) positions with incomplete occupancies. The plane. The Mn-octahedra are hatched. Circles of different ﬁnal atomic coordinates are given in Table 2. The crys- types represent large cations and water molecules.

230 ROZENBERG et al.

Table 1. Structural data and details of collection of the X-ray data Characteristic Parameter Characteristic Parameter Parameters of monoclinic unit cell, Å, deg a = 14.298(7) Ranges of the indices of measured reflections Ð16 < h < 14; b = 13.816(7) Ð15 < k < 16; c = 7.792(3) 0 < l < 9 β = 116.85(5)° sinθ/λ <0.58 Volume of monoclinic unit cell, Å3 1373.3 Total number of reflections 1235 Sp. gr.; ZC2/m; 8 Number of independent reflections 1084 F > 4σ(F) Radiation; λ, Å CuKα; 1.5411 R-factor upon anisotropic refinement 0.047 Density, g/cm3 2.9 Program used for the refinement AREN [2] Crystal dimensions, mm 0.15 × 0.1 × 0.25 Program used for introduction of the DIFABS [3] absorption correction

Diffractometer SYNTEX P21

Table 2. Atomic coordinates, equivalent thermal parameters, multiplicities, and occupancies of the positions

2 Atom x/ay/bz/cQ qBeq, Å Si(1) 0.2082 (1) 0.1102 (1) 0.8035 (1) 8 1 1.18 (7) Si(2) 0.3199 (1) 0.1111 (1) 0.2493 (1) 8 1 1.25 (7) Ti(1) 0 0.2276 (1) 0.5 4 1 1.55 (6) Ti(2) 0.25 0.25 0.5 4 1 1.66 (6) D 0 0 0.5 2 0.47(1) 1.6 (1) C 0.873 (1) 0 0.3422 (1) 4 0.83(1) 2.16 (3) K(1) 0.4207 (1) 0 0.7033 (3) 4 0.69(1) 2.7 (1) K(2) 0.4246 (8) 0.0444 (8) 0.713 (2) 8 0.15(1) 4.5 (3) Na 0.4130 (4) 0.2643 (3) 0.0077 (9) 8 0.44(1) 3.4 (2) O(1) 0.2336 (2) 0.1278 (2) 0.272 (4) 8 1 1.8 (2) O(2) 0.2736 (2) 0.1823 (2) 0.7377 (4) 8 1 1.8 (2) O(3) 0.2461 (3) 0 0.7847 (7) 4 1 1.8 (3) O(4) 0.4204 (2) 0.1802 (2) 0.3028 (4) 8 1 1.9 (2) O(5) 0.2649 (2) 0.1260 (2) 0.3864 (4) 8 1 1.6 (2) O(6) 0.3640 (3) 0 0.2708 (6) 4 1 1.7 (3) O(7) 0.832 (2) 0.1169 (2) 0.6741 (4) 8 1 1.7 (2) O, OH 0.1014 (2) 0.2255 (2) 0.3974 (4) 8 1 1.7 (2)

H2O (1) 0.5 0.1468 (8) 0 4 0.68(2) 8.5 (3)

H2O (2) 0 0.1125 (6) 0 4 0.68(2) 5.2 (2)

Note: D = Mn0.75 + Fe0.25; C = Ba0.45 + (H2O)0.55; Ti(2) = Ti0.95 + Nb0.05.

columns of vertex-sharing [TiO6]-octahedra linked by cule and a Ba atom occupy the mixed C-position due to four-membered rings of [SiO4]-tetrahedra. The chan- which the coordination polyhedron of a Mn atom is nels in the structure are occupied by Na, K, and Ba complemented to an octahedron, as opposed to the atoms and water molecules. The chemical composition structure of lemmleinite-Ba, in which H2O molecules and the structure of the new specimen make it similar to and Ba atoms occupy different although close posi- lemmleinite-Ba; however, there are also some differ- tions. Whereas in the structure of lemmleinite-Ba, K ences. Like other labuntsovites, the H2O(1) and H2O(2) atoms are located in one position. In the structure of the molecules in the new mineral are in the coordination specimen studied, these atoms occupy two positions spheres of Na atoms, with each water molecule being spaced by a distances of 0.61(1) Å, with the position bound to one of the large cations. The third H2O mole- K(2) being occupied by 15%.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

STRUCTURAL CHARACTERISTICS OF A NEW CATION-DEFICIENT REPRESENTATIVE 231

According to the nomenclature of the Commission ACKNOWLEDGMENTS on New Minerals and Mineral Names of the Interna- This study was supported by the Russian Founda- tional Mineralogical Association, four positions in the tion for Basic Research, project no. 99-05-65035. channels of the crystal structure are group-forming ones in the minerals under consideration. Two of these positions, namely, the D position occupied by medium- REFERENCES sized cations and the C position occupied by large cat- 1. R. K. Rastsvetaeva, N. V. Chukanov, and I. V. Pekov, ions (in some earlier studies, these positions were indi- Dokl. Akad. Nauk 357 (1), 64 (1997). cated as MII and AIII, respectively) complement each 2. V. I. Andrianov, Kristallograﬁya 32, 228 (1987) [Sov. other statistically. In the structure of the new mineral, Phys. Crystallogr. 32, 130 (1987)]. the D position is occupied mainly by Mn atoms (47%), whereas the C position is occupied by Ba-atoms (37%) 3. N. Walker and D. Stuart, Acta Crystallogr., Sect. A: Found. Crystallogr. A39 (2), 158 (1983). and water molecules (47%). Hence, the cationic vacancies prevail in both positions, which distinguishes the new mineral from other representatives of this group. Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

STRUCTURE OF INORGANIC COMPOUNDS

Structural Characteristics of Na,Fe-Decationated Eudialyte with the Symmetry R3 R. K. Rastsvetaeva* and A. P. Khomyakov** * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia e-mail: [email protected] ** Institute of Mineralogy, Geochemistry, and Crystal Chemistry of Rare Elements, ul. Veresaeva 15, Moscow, 121357 Russia Received March 15, 2001

Abstract—The crystal structure of a new highly decationated representative of the eudialyte group has been established (R = 0.055, 1734 |F|). The mineral is described by the simpliﬁed formula (H3O)9Na2(K,Ba,Sr)2Ca6Zr3[Si26O66(OH)6](OH)3Cl á H2O (Z = 3). The unit-cell parameters are a = 14.078(3) Å, c = 31.24(1) Å; V = 5362 Å3; sp. gr. R3. Being chemically and structurally related to the hydrated analogues studied previously (in particular, to potassium oxonium eudialyte), the new mineral differs from its analogues in that it has a higher degree of Na- and Fe-cation depletion. The replacement of 3/4 of Na cations by loose and mobile H3O groups results in structure destabilization, which is seen from the high values of the thermal parameters of the atoms and the loss of the symmetry plane. © 2002 MAIK “Nauka/Interperiodica”.

The properties of zeolite-like zirconosilicate eudia- tions of all framework atoms of the mineral were lyte as a potential ion exchanger are clearly seen from revealed from a series of Fourier syntheses. At the final the existence of its decationated analogues character- stage, the positions with low occupancies were localized by high degrees of hydration. Earlier, we estab- ized from difference electron density synthesis. The lished the structures of two such specimens containing inclusion of the impurity atoms with the use of mixed up to 7 wt % of water, which were found in the Lovoz- atomic scattering curves led to a sufficiently low R fac- ero and Khibiny massifs [1, 2]. In this study, we examined the eudialyte-like mineral found in syenite pegma- tites from the Inaglinskii massif (South Yakutia). This Table 1. Structural data and details of X-ray diffraction mineral is characterized by a higher degree of hydra- study of the crystal tion (up to 10 wt % of H2O) and by a very low (about 3.5 wt %) total content of Na and Fe oxides. According Unit-cell parameters, Å a = 14.078(3), c = 31.24(1) to Efimov et al. [3], this fact is responsible for the pale- Unit-cell volume, Å3 V = 5362 3 ρ ρ pink color of the mineral, which is rather rare for widely Density, g/cm : calcd; obs 2.7; 2.62 occurring eudialytes, the low density (2.62 g/cm3), and the Sp. gr., ZR3; 3 (for the simplified low values of the refractive indices (1.567Ð1.572). formula) The results of microprobe analysis together with the Radiation, λ, Å MoKα; 0.71073 data on the water content determined by the Penfield Crystal dimensions, mm 0.15 × 0.25 × 0.2 method corresponds to the empirical formula (Z = 3) Diffractometer Enraf-Nonius Na K Sr Ba Ca 2.74 1.20 0.49 0.46 5.79 sinθ/λ <0.7 á REE0.19Fe0.23Mn0.12Zr2.92Ti0.29Nb0.03Al0.19Si25.57 O66.6Cl1.23 á nH O Ranges of the indices of Ð19 < h < 17, 0 < k < 19, 2 measured reflections 0 < l < 42 (where n is ~16). R or equivalent reflections 0.029 X-ray diffraction data were collected from an iso- av σ metric single crystal. The characteristics of the single Total number of reflections 2790 I > 2 (I) crystal and the details of the X-ray diffraction study are Number of independent reflections 1734 |F| > 4σ(F) given in Table 1. Program for calculations AREN[4] Based on the similarity of the compositions of the Program for absorption correction DIFABS [5] mineral under study and potassium oxonium eudialyte Number of independent positions 60 [1], we assumed that they are structurally similar and used the coordinates of the positions of the framework R factor upon anisotropic 0.055 atoms of this analogue as the starting model. The posi- refinement

STRUCTURAL CHARACTERISTICS OF Na,Fe-DECATIONATED EUDIALYTE 233 tor. However, the thermal parameters of all the atoms Table 2. Coordinates and equivalent (Beq) thermal parame- were too high. Although the refinement within the ters of the framework atoms sp. gr. R3 did not lead to a substantial improvement in 2 the R factor, it gave reliable thermal parameters. The Atom x/ay/bz/cBeq, Å lowering of the symmetry compared to the sp. gr. R3m, characteristic of a number of structures of the eudialyte Zr 0.3352(2) 0.1681(2) 0.1669(1) 2.48(3) group, is most often associated with the ordering of the Ca 0.4018(2) 0.3312(2) 0.3330(1) 2.25(5) Ca atoms and of some elements replacing these atoms in the positions of the six-membered rings. In the min- Ca' 0.4011(2) 0.0702(2) 0.3332(1) 2.45(5) eral under study, the Ca and (Ca,Ce) atoms are ordered. Si(1) 0.5236(2) 0.2616(2) 0.2529(1) 1.4(1) Apparently, the lowering of the symmetry is associated mainly with the destabilization of the structure caused Si(2) Ð0.0088(3) 0.6029(3) 0.1014(1) 3.1(1) by the replacement of Na atoms by oxonium groups, Si(2)' Ð0.0097(3) 0.3880(3) 0.1006(1) 2.4(1) which results in systematic displacements of the atoms from the vertical plane m exceeding the experimental Si(3) 0.2033(3) 0.4077(3) 0.0790(1) 2.9(1) error. The most substantial displacements are observed Si(4) 0.0828(3) 0.5411(3) 0.2595(1) 2.3(1) for the oxonium groups characterized by the highest mobility and located in their own subpositions with Si(5) 0.0511(3) 0.3223(4) 0.2360(1) 2.5(1) occupancies less than 100%. Si(5)' 0.2726(3) 0.3233(4) 0.2354(1) 3.2(1) The final atomic coordinates are given in Tables 2 and 3. Some characteristics of the polyhedra are indi- Si(6) 0.1411(3) 0.0709(3) 0.0816(1) 2.1(1) cated in Table 4. O(1) 0.4596(9) 0.2336(11) 0.2079(4) 3.2(3) The principal characteristics of the composition and the structure of hydrated eudialyte are described O(2) 0.2491(10) 0.0218(10) 0.2022(4) 3.2(4) by its crystallochemical formula (Z = 1) O(2)' 0.2547(10) 0.2267(8) 0.2050(4) 3.2(3) []VI Zr9(Ca17.64Ce0.36)[Si72O198.7(OH)17.3][Si3][Si1.5 Ti0.75 á O(3) 0.4082(10) 0.3108(10) 0.1339(4) 3.5(3) [] [] [] [VI] IX IV VI (Al0.65Nb0.1) ][(H3O)4.4 Na2.2 (Fe,Mn)1.0 ][(H3O)20.9 á O(3)' 0.4121(8) 0.0962(9) 0.1362(3) 2.6(3) Na6.5K3.6Ba1.6Sr1.31Ce0.2][(OH)9.6Cl3.9(H2O)2.16(SO4)0.6], O(4) 0.6064(8) 0.3966(8) 0.2512(5) 3.7(3) where the compositions of the key structural fragments are indicated in brackets and the coordination numbers O(5) 0.4542(9) 0.2253(10) 0.2961(4) 3.3(3) of the cations are given by Roman numerals in brackets. O(6) 0.4015(14) 0.0570(16) 0.0520(5) 5.7(4) The sulfate group established by the X-ray diffraction analysis replaces the Cl atom or the H2O molecule O(6)' 0.3974(10) 0.3470(11) 0.0532(4) 4.4(4) occupying the corresponding position on a threefold O(7) 0.0990(9) 0.3736(10) 0.1063(4) 3.2(3) axis in a number of other minerals of the eudialyte group.1 O(7)' 0.2773(6) 0.3722(6) 0.1062(3) 1.6(2) The simplified formula of the mineral is O(8) 0.0215(9) 0.5123(8) 0.1071(4) 2.9(3) ⋅ (H3O)9Na2(K,Ba,Sr)2Ca6Zr3[Si26O66(OH)6](OH)3Cl O(9) 0.2710(10) 0.5480(8) 0.0739(5) 4.0(3) H2O (Z = 3). The mineral is characterized by a high oxonium O(10) 0.1850(10) 0.3584(10) 0.0265(4) 3.7(3) content. The X-ray diffraction analysis provided only O(11) 0.0242(10) 0.5171(8) 0.3024(4) 3.2(3) indirect evidence of the presence of oxonium groups, because only O atoms were localized in the Na posi- O(12) 0.1817(10) 0.3616(10) 0.2205(4) 3.3(4) tions from electron density maps. We assigned these O + O(13) 0.0370(16) 0.2918(16) 0.2866(5) 5.7(4) atoms to (H3O) ions. Taking into account a substantial deficiency in positive charges, these groups cannot be O(13)' 0.2589(11) 0.2909(12) 0.2835(4) 5.3(4) interpreted as neutral H2O molecules. The presence of O(14) 0.3904(10) 0.4279(11) 0.2302(5) 3.9(4) oxonium groups was also confirmed by the IR spectra of hydrated eudialytes [2]. Compared to other oxo- O(14)' 0.3946(15) Ð0.0381(17) 0.2293(6) 6.5(3) nium-containing minerals [1, 2], this specimen has the O(15) 0.3978(14) 0.6073(14) 0.2557(5) 5.1(1) maximum amount of H3O groups. In addition to three Na positions, in which oxonium groups were revealed O(16) 0.0625(9) 0.1262(8) 0.0817(4) 2.8(2) earlier, we established the Na(2) position and two posi- O(17) 0.2006(9) 0.1038(9) 0.1247(3) 2.9(2) tions in the plane of the Ca-rings (figure). In high-alka- O(18) 0.2158(10) 0.1052(10) 0.0424(3) 3.1(3) 1 Sulfur was not detected in the grains studied by the microprobe analysis. Note: The atoms related by the pseudosymmetry plane m are primed.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 234 RASTSVETAEVA, KHOMYAKOV

Table 3. Coordinates and equivalent thermal parameters (Beq) of the extraframework atoms, multiplicities (Q), and occupancies (q) of the positions

2 Atom x/ay/bz/cQ q Beq, Å Si(7a) 0.3333 0.6667 0.2773(6) 3 0.55(3) 5.6(4) Si(7b) 0.3333 0.6667 0.2449(5) 3 0.45(3) 2.45(3) Si(8) 0.3333 0.6667 0.0600(6) 3 0.50(2) 2.5(3) Ti 0.3333 0.6667 0.1009(9) 3 0.25(1) 3.7(5) Al 0.3333 0.6667 0.044(2) 3 0.25(2) 6.3(5) S 0.6667 0.3333 0.040(3) 3 0.20(4) 6.9(5) M 0.492(1) 0.510(1) Ð0.0004(7) 9 0.35(1) 3.0(3) A(1) 0.1153(7) 0.2265(8) 0.1519(3) 9 1.0 2.2(3) A(2a) 0.5631(3) 0.4360(3) 0.1792(1) 9 0.71(1) 2.7(1) A(2b) 0.565(3) 0.436(3) 0.1648(9) 9 0.29(4) 7.5(7) A(3a) 0.192(1) 0.096(2) 0.2942(6) 9 0.60(1) 6.7(3) A(3b) 0.158(1) 0.079(1) 0.2805(4) 9 0.40(4) 5.5(3) A(4a) 0.464(2) 0.232(2) 0.0501(4) 9 0.30(4) 5.7(4) A(4b) 0.534(2) 0.266(2) 0.045(1) 9 0.70(1) 8.0(5) A(5a) 0.224(3) 0.464(5) 0.322(1) 9 0.32(4) 4.2(5) A(5b) 0.416(3) 0.596(2) 0.005(1) 9 0.39(4) 3.8(8) OH(1) 0.3333 0.6667 0.187(1) 3 0.67(6) 3.5(6) OH(2) 0.222(4) 0.604(4) 0.162(1) 9 0.27(4) 5.6(6) OH(3) 0.3333 0.6667 0.331(1) 3 0.55(9) 7.9(5) OH(4) 0.3333 0.6667 0.005(1) 3 0.50(3) 8.5(6) Cl(1) 0 0 0.197(1) 3 0.50(3) 6.7(6) Cl(2a) 0.6667 0.3333 0.1424(4) 3 0.59(5) 4.4(4) Cl(2b) 0.6667 0.3333 0.098(2) 3 0.20(3) 6.9(5)

H2O 0.631(5) 0.380(5) 0.132(2) 9 0.24(4) 6.1(7) Note: Hereafter, M denotes the position in the center of the “square”; A(1Ð5) are the cationic positions corresponding to the Na(1Ð5) positions in the structures of high-sodium eudialytes. line analogues of eudialyte, the latter positions are other two minerals, the Na-, and Zr-octahedra are ori- occupied either by Na [6] or K [7]. The distance ented toward the cavities [1, 2]. between the positions of these cations correlates with the cation sizes and increases from the minimum value It can be assumed that the presence of oxonium ions in the Ca layer is responsible for the noticeable elonga- (2.8 Å) in alluaivite [6] to the maximum value (3.8 Å) tion of the structure in the direction of the threefold in the new mineral. In the structure under consider- axis. Thus, the c-parameter of the unit-cell (31.24 Å) ation, the Na(1) position is fully occupied by the oxo- considerably exceeds the c-parameter in typical eudia- nium groups (as in the specimen found in Khibiny); the lytes where it varies from 29.96 to 30.35 Å [8]. The remaining Na positions, except for Na(2), are also lower density of the mineral is most likely associated occupied predominantly by oxonium groups. with the more flexible Ca-containing layers and not with the rigidly fixed fragments (Zr,Si-containing lay- In all three structures, the cavities between the rings ers), in particular, with the absence of the Fe atoms in are filled with hydroxyl groups forming a tetrahedral the former layers. This assumption was confirmed by and octahedral environment around the cations located the distortion of the “square.” The edges of the Ca-octa- on threefold axes. The octahedra located on the axes are hedra forming two opposite sides of the square (3.32 oriented in the direction of the cavities. Such an orien- and 3.42 Å) are substantially longer than those found tation is characteristic only of hydrated specimens and earlier (2.79–3.24 Å). The elongation of the Ca-octahe- is associated with the absence of Na atoms in these cav- dra along the z-axis is accompanied by their contraction ities. Indeed, in the new mineral, the Ti-octahedron is in the perpendicular direction, which provides the min- oriented in the direction of the cavity, whereas in the imum value of the a-parameter of the unit-cell (14.08 Å)

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STRUCTURAL CHARACTERISTICS OF Na,Fe-DECATIONATED EUDIALYTE 235

Table 4. Characteristics of the coordination polyhedra

Coordination CationÐanion distances Position Composition, Z = 1 number limiting average Zr 9Zr 6 1.98Ð2.11 2.06 Ca 9Ca 6 2.25Ð2.40 2.33 Ca' 8.64Ca + 0.36Ce 6 2.24Ð2.46 2.35 Si(7a) 1.65Si 4 1.64Ð1.68 1.65 Si(7b) 1.35Si 4 1.55Ð1.80 1.61 Si(8) 1.5Si 4 1.51Ð1.72 1.55 Ti 0.75Ti 6 1.67Ð2.34 2.01 Al 0.65Al + 0.1Nb 6 1.72Ð2.23 1.97 S 0.6S 4 1.31Ð1.62 1.54 M 2.2Na + 1.0(Fe, Mn) 4 2.11Ð2.60 2.39 6 1.85Ð2.60 2.23 A(1) 9H3O 9 2.49Ð3.10 2.67 A(2a) 4.7Na + 1.6Ba 9 2.37Ð2.88 2.66 A(2b) 2.5H3O + 0.2Ce 7 2.14Ð2.88 2.59 A(3a) 4.9H3O + 0.5Sr 9 2.44Ð3.28 2.78 A(3b) 3.6K 9 2.58Ð3.22 2.85 A(4a) 1.8Na + 0.81Sr 8 2.16Ð3.17 2.80 A(4b) 4.5H3O 9 2.19Ð3.20 2.85 A(5a) 1.9H3O + 1.0OH 9 2.49Ð3.31 2.89 A(5b) 2.5H3O + 1.0OH 7 2.58Ð3.36 2.91 Note: The Si(1Ð6)-tetrahedra involved in the siliconÐoxygen rings of the framework are not indicated. with a simultaneous increase in the distance between Therefore, the new mineral is related, both chemi- the rings (the lengths of the two other sides of the cally and structurally, to the hydrated analogues studied square are 3.58 and 3.13 Å; the corresponding values in earlier (and, in particular, is similar to potassium oxo- other known eudialytes are in the range 2.80Ð3.04 and nium eudialyte) and is characterized by the highest 2.91–3.25 Å, respectively). degree of depletion of Na- and Fe-cations. The struc-

A layer of the decationated eudialyte structure projected onto the (001) plane; Si-tetrahedra are hatched with solid lines; Na atoms and oxonium cations are indicated by small and large circles, respectively.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 236 RASTSVETAEVA, KHOMYAKOV ture of the new mineral is less stable because of the 2. I. A. Ekimenkova, R. K. Rastsvetaeva, and N. V. Chu- replacement of 3/4 of the Na atoms by looser and kanov, Dokl. Akad. Nauk 371 (5), 625 (2000). mobile H3O groups. This is seen from the high thermal 3. A. F. Efimov, S. M. Kravchenko, and E. V. Vlasova, Tr. parameters of the atoms, the loss of the symmetry IMGRÉ 16, 141 (1963). plane, the distortion of the coordination polyhedra, and the shortening of some interatomic distances. In the 4. V. I. Andrianov, Kristallografiya 32 (1), 228 (1987) [Sov. Phys. Crystallogr. 32, 130 (1987)]. new mineral, the characteristic splitting of positions of the extraframework atoms into partly occupied subpo- 5. N. Walker and D. Stuart, Acta Crystallogr., Sect. A: sitions characterized by isomorphous replacement is Found. Crystallogr. A39 (2), 158 (1983). most pronounced. 6. R. K. Rastsvetaeva, A. P. Khomyakov, V. I. Andrianov, and A. I. Gusev, Dokl. Akad. Nauk SSSR 312 (6), 1379 (1990) [Sov. Phys. Dokl. 35, 492 (1990)]. ACKNOWLEDGMENTS 7. R. K. Rastsvetaeva and A. P. Khomyakov, Kristal- This study was supported by the Russian Founda- lografiya 46 (4), 715 (2001) [Crystallogr. Rep. 46, 647 tion for Basic Research, project no. 99-05-65035. (2001)]. 8. O. Johnsen and J. D. Grice, Can. Mineral. 37 (4), 865 REFERENCES (1999). 1. R. K. Rastsvetaeva, M. N. Sokolova, and B. E. Borutskiœ, Kristallografiya 35 (6), 1381 (1990) [Sov. Phys. Crystal- logr. 35, 814 (1990)]. Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 237Ð244. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 272Ð279. Original Russian Text Copyright © 2002 by Arakcheeva, Grinevich, Chapuis, Shamraœ. STRUCTURE OF INORGANIC COMPOUNDS

Structure Studies of Solid Solutions of Oxygen in Electrolytic Niobium A. V. Arakcheeva*, V. V. Grinevich*, G. Chapuis**, and V. F. Shamraœ* * Baikov Institute of Metallurgy and Materials Technology, Russian Academy of Sciences, Leninskiœ pr. 49, Moscow, 117334 Russia e-mail: [email protected] ** Institute of Crystallography, University of Lausanne, BSP, Lausanne, 1015 Switzerland Received April 17, 2001

Abstract—The crystal structures (cubic system, sp. gr. Im3m) of solid solutions of oxygen in niobium of the compositions NbOx < 0.01 (I), NbO0.150(6) (II) and NbO0.107(5) (III) were studied for the ﬁrst time by single-crystal X-ray diffraction analysis (Syntex P1 diffractometer, λMoKα radiation). Crystals I (a = 3.2969(8) Å) and II (a = 3.3082(6) Å) were prepared by the electrolysis of salt melts. Crystal III (a = 3.3114(4) Å) was obtained after partial dissolution of crystal I in a solution of HNO3, HCl, and HF resulting in its saturation with oxygen. The residual electron density maps for crystal I revealed no O atoms. Localization of the O atoms in the tetrahedral cavities of crystal II and in the tetrahedral and octahedral cavities of crystal III indicates that the arrangement of the interstitial positions of oxygen atoms depends on the procedure of preparing solid solutions. In crystals II and III, the amplitudes of thermal vibrations of the Nb atoms are smaller and the anharmonicity of these vibrations is higher than those in crystal I virtually devoid of oxygen. The reasons for the substantially higher oxygen content (up to 17 at. %) in cubic crystals of the solid solutions of O in Nb prepared by the electrolytic method than in the analogous phase obtained upon heating (lower than 7 at. %) are discussed. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION quantum-mechanical analysis, the tetrahedral and octa- Interactions of oxygen with niobium metal are being hedral cavities in the body-centered cubic (bcc) struc- extensively studied in connection with the problem of ture of Nb were considered as positions occupied by the corrosion resistance of metal at high temperatures oxygen atoms [13, 14]. The dependence of the mecha- and the influence of oxygen on the mechanical proper- nisms of incorporation of oxygen atoms into the struc- ties of niobium [1]. Interest in these studies stems from ture of niobium metal on the mode of preparation of the the fact that niobium saturated with oxygen is used as a solid solution was considered in the study of Nb sam- collector in thermionic energy converters [2, 3]. It was ples, which were saturated with oxygen either in the established [1–8] that the first stage of the reaction of course of gas-phase reactions or by thermal diffusion oxygen with niobium gives rise to solid solutions with saturation [12]. However, the preparation of homoge- an increased cubic unit-cell parameter, followed by the neous samples of cubic solid solutions of oxygen in formation of the suboxide phases Nb6O and Nb4O char- niobium in the case of a low oxygen content in equilib- acterized by the tetragonal distortion of the cubic unit rium phases (0.7 at. % at 700°C and 5.2 at. % at 1500°C cell, after which stoichiometric oxides are generated. It [8]; lower than 1 at. % at 550°C and ~7 at. % at 1700°C should be noted that the tetragonal suboxide phase, [15]) presented difficulties, because they were formed which has been initially identified as Nb2O [9], was as thin interlayers between oxides and the metal upon heating of niobium in air. As a result, the positions of demonstrated to be stoichiometric oxide Nb4O5 [10] possessing a unique structure [11]. It was shown [12] the oxygen atoms were impossible to localize by X- that the formation of solid solutions of oxygen in nio- ray diffraction methods. We prepared well-faceted bium substantially affects the physical properties of homogeneous single crystals of cubic solid solutions lower oxides. In particular, two lower oxides with with oxygen contents up to NbO0.17 by electrolytic NaCl-type structures were described [12], among them crystallization from salt melts and then saturated these the well-known NbO phase with the narrow homogene- single crystals with oxygen upon etching in a ity region and the NbO1.2 phase possessing high hard- (HNO3 + HCl + HF) solution, which allowed us to ness. The characteristic features of the oxygen arrange- localize O atoms in the Nb structure by single-crystal ment in the lattice of niobium metal were discussed in X-ray diffraction methods. Below, we present the several studies [12Ð14]. According to the results of results of this study.

238 ARAKCHEEVA et al. PREPARATION OF SAMPLES cycles of electrolysis. Samples with x < 0.17 were Samples of niobium with a controlled oxygen con- obtained as well-faceted crystals with an approximately tent were prepared by electrolysis of a salt melt of com- cubic habit. X-ray phase analysis (DRON-3M diffractometer, λMoKα radiation) showed that the samples position KCl + NaCl + K2NbF7 (20 wt %) using a soluble niobium-monoxide anode and a molybdenum were single-phase and demonstrated that they have the cathode. cubic symmetry (sp. gr. Im3m) typical of the body-centered cubic structure of Nb. Recrystallized potassium heptafluoroniobate and alkali-metal chlorides of special purity grade were used Single crystals with the minimum and maximum as components of the electrolyte. The niobium-monox- oxygen contents of compositions NbOx < 0.01 (I) and ide anode was prepared by arc melting pressed pellets NbO0.17(1) (II), respectively, were studied by X-ray dif- using a permanent electrode. The pellets were made fraction analysis. Several crystals I were soaked in a from a mixture of highly pure electrolytic niobium pre- dilute aqueous solution (1 : 1 v/v) of HNO3 + HCl + HF pared according to a known procedure [16] and Nb2O5 for four hours to reduce their rather large sizes taken in a molar ratio of 3 : 1. (~0.3 mm) by partial dissolution. Due to the presence A cylindrical molybdenum crucible with a cylindri- of the strong oxidizer, the crystals were not only cal perforated molybdenum diaphragm inside the cru- decreased in size, but also enriched with oxygen, and cible was used as the electrolytic cell. The anodic mate- the resulting samples (III) were also studied by X-ray rial (NbO pieces of size 3Ð8 mm) was charged into the analysis. As a result, we obtained data on the structure circular space between the diaphragm and the walls of of the solid solution of O in Nb, which was prepared by the crucible. A molybdenum rod with diameter 5 mm, the “chemical” method. which was fixed coaxially in the center of the cell, served as the cathode. The cell was placed in a heat- resistant steel container, in which a highly pure atmo- X-RAY DIFFRACTION STUDY sphere of argon was maintained. The electrolytic cell was heated externally in a shaft furnace with silicon- Single crystals I, II, and III selected for X-ray dif- carbide heaters. fraction study had the shapes of isometric parallelepi- peds with virtually perfect mirror-reflecting rectangular Electrolysis was carried out in the galvanostatic 2 faces. The a parameters of the cubic unit cells were mode with the cathode-current density of 0.2 A/cm at measured on a Syntex P diffractometer (λ Kα radi- ° 1 Mo 750 C. In each cycle, a direct current of 1.5 A h was ation) based on 15 high-angle reflections of the 〈224〉 passed through the electrolytic cell. type lying in the 2θ angle range corresponding to λ The cathode with a precipitate grown on its surface MoKα1 = 0.70926 Å (Table 1). In none of the three during one cycle of electrolysis was cooled to room crystals were significant deviations from the cubic metric temperature in a water-cooled magazine of the con- revealed. The a parameter of crystal I (a = 3.2969(8) Å) tainer cover. The crystalline precipitate of the electrol- agrees well with the tabulated value for Nb (a = ysis product was separated from the cathode rod, 3.2986 Å) [19] and is in the range 3.294–3.300 Å cor- washed off from the salt electrolyte, which was cap- responding to the cubic unit-cell parameters for Nb tured from the electrolytic cell, with 10% hydrochloric containing no oxygen [1Ð9, 12Ð14]. The largest param- acid, washed repeatedly with distilled water and then eter a = 3.3114(4) Å was found for crystal III. This fact with rectified ethanol, and dried in air at 30–40°C. indicates that crystal III, which was obtained from the The electrochemical and chemical (due to interactions sample with the lowest oxygen content by partial disso- with the NbF 2Ð ions) dissolution of NbO in a melt gave lution in a solution of HNO3, HCl, and HF, was sub- 7 stantially saturated with oxygen. Since we had only a Ð 3Ð rise to complex oxofluoride ions NbOF4 and NbOF6 few such crystals with an edge size of ~0.1 mm at our [17], whose concentrations in the electrolyte increased disposal, we did not determine the oxygen content in with the increasing total quantity of the direct current crystals III by high-temperature oxidation. passed through the electrolytic cell. Since the strict cor- The principal details of the X-ray diffraction studies relation between the O : Nb molar ratio in the molten and the characteristics of the crystals are given in electrolyte and the oxygen content in the cathodic prod- Table 2. For all three crystals, the experimental inte- uct was observed in the electrolysis of fluoride–chlo- grated intensities of X-ray diffraction reflections (2θ/θ ride melts [18], cathodic precipitates obtained in suc- scanning technique) were measured within a hemi- cessive cycles of electrolysis contained increasing con- sphere of the reciprocal space. All experimental reflec- centrations of oxygen. tions were characterized by sharp profiles, which is The concentration of oxygen in each cathodic pre- indicative of the high quality of the single crystals. All cipitate was determined from the increase in the weight reciprocal-lattice vectors involved in the measured of the starting sample after its complete high-tempera- region were scanned. The intensities of all the reflec- ture oxidation in air to Nb2O5. A series of NbOx sam- tions allowed within the space group Im3m were higher ples with 0 < x < 1.32 were prepared in successive than 3σ(I). All the reflections were consistent with this

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STRUCTURE STUDIES OF SOLID SOLUTIONS 239

Table 1. Compositions, unit-cell parameters (a), and positions of the oxygen atoms in NbOx crystals I, II, and III (sp. gr. Im3m) Crystal I II III Composition determined by

the oxidation method NbOx < 0.01 NbO0.17(1) Was not determined

from the results of the refinement Nb NbO0.150(6) NbO0.107(5) θ λ 〈 〉 2 angle range (deg) for Mo Kα1 for the 224 63.58Ð63.65 63.34Ð63.37 63.27Ð63.30 reflections used in the refinement of the a parameter a, Å 3.2969(8) 3.3082(6) 3.3114(4) Wyckoff position and configurations of the cavities O atoms were 12d, tetrahedron 12d, tetrahedron; occupied by O atoms not found 6b octahedron

Table 2. Details of X-ray diffraction studies and results of the reﬁnement of the Nb (I) and NbOx (II and III) structures within the space group Im3m Crystal I II III

Composition Nb NbO0.150(6) NbO0.107(5) Dimensions, mm 0.25 × 0.25 × 0.20 0.20 × 0.20 × 0.22 0.10 × 0.12 × 0.12

Diffractometer; radiation Syntex P1; MoKα N ; N (sinθ/λ < 1.07); 429; 31; obs indep θ λ Nindep (0.7 < sin / < 1.07) 20 Absorption Integration with respect to the real habit of the crystal

Rint for equivalent reflections 0.033 0.025 0.022 Reﬁnement with isotropic thermal parameters of Nb atoms [2a: (000)] 2 Nb: Biso, Å 0.89(1) 0.289(1) 0.306(7) θ λ RF; wRF (0 < sin / < 1.07) 0.0188; 0.0054 0.0122; 0.0093 0.0125; 0.0067 θ λ RF: wRF (0.7 < sin / < 1.07) 0.0296; 0.0286 0.0083; 0.0074 0.0092; 0.0077 ∆ρ ∆ρ max; min or deformation electron 3.07; Ð1.76 0.94; Ð0.52 1.06; Ð0.48 density, e/Å3 Reﬁnement of thermal parameters of Nb atoms in the anharmonic approximation 2 Nb: Beq, Å 0.54(1) 0.224(3) 0.262(6) B11 0.01237(1) 0.0051(1) 0.0060(2) D1111 × 104 Ð0.002(1) Ð0.20(2) Ð0.027(3) D1122 × 104 Ð0.0031(7) Ð0.0069(7) Ð0.006(1) F111111 × 106 Ð0.010(3) Ð0.027(3) Ð0.035(5) F111122 × 106 0.0008(8) Ð0.0046(6) Ð0.004(1) F112233 × 106 0.001(1) Ð0.0014(4) 0.0000(7)

RF; wRF 0.0092; 0.0049 0.0087; 0.0086 0.0083; 0.0055 ∆ρ ∆ρ 3 max; min, e/Å 0.61; Ð0.60 0.69; Ð0.61 0.80; Ð0.60 Localization of O atoms 2 O(1) [12d: (0.5 0.25 0)]: Biso, Å Ð 2.2(2) 0.49(8) q 0.025(1) 0.0089(3) 2 O(2) [6b: (0.5 0.5 0)]: Biso, Å Ð 2.1(7) q 0.018(1)

RF; wRF 0.0072; 0.0041 0.0061; 0.0031 ∆ρ ∆ρ 3 max; min, e/Å 0.35; Ð0.48 0.30; Ð0.35

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 240 ARAKCHEEVA et al. space group that characterized the bcc structure of nio- (a) bium metal. x = 0 All calculations were carried out with the use of the JANA98 program package. The structure characteristics for all three crystals were determined and refined 0.8 according to the same scheme. The absorption corrections were applied by numerical integration with respect to the real habit of the crystals. Then the intensities of X-ray reflections were converted into structure factors taking into account the Lorentz and polarization factors. The parameters of each structure were refined using the same number of structure factors (31), which 0.4 were obtained by averaging the equivalent reflections of the same hkl set. Cycles of the refinement by the full- matrix least-squares method based on |F| were alter- nated with cycles of the refinement based on F 2 . The scale factor and the isotropic thermal parameter of the Nb atom were refined using all independent reflections 0 with 0 < sinθ/λ < 1.07. Then, B of the Nb atom was (b) iso refined based on high-angle reflections with 0.7 < sinθ/λ < 1.07. The deformation electron density maps, which were constructed based on the refined data (Fig. 1), revealed substantial dynamic (anharmonicity) or static 0.8 (splitting of the position) thermal displacements of the Nb atoms along the coordination axes. This effect was adequately parametrized using the sixth-order GramÐ Charlier expansion in series for the anharmonicity of

c thermal atomic displacements (4.43 reflections per / z parameter to be refined). Each probability density func- 0.4 tion for the displacements of the Nb atoms (Fig. 2) has one maximum elongated along the coordination axes. A comparative analysis of the difference electron density maps for crystals I, II, and III (Fig. 3) revealed information about the positions of the oxygen atoms in crystals II and I and provided evidence that crystals I 0 contained no oxygen. For each oxygen position, the (c) isotropic thermal parameter Biso and the occupancy q were determined by minimizing the R factor as a function of these two correlating parameters. The R values were obtained by refining the occupancies q at fixed 0.8 values of the Biso parameter, which was varied from 0.2 to 3.0 with a step of 0.1. Thereafter, the residual electron density maps were constructed for crystals II and III (Fig. 4). All numerical characteristics of the refinement obtained at each stage for crystals I, II, and III are given in Table 2. (All results were obtained using all 0.4 independent reflections, except for the results of the refinement based on high-angle reflections, which are presented in the individual line.) Let us consider the characteristic features of each crystal in more detail. Compared to crystals II and III, crystal I (Table 2) is characterized by the maximum size (Vcryst = 3 0 0.0125 mm ) and the highest R factor of averaging of 0 0.4 0.8 the symmetrically equivalent reflections (Rint = 0.033), y/b which is, apparently, responsible both for the highest final R factors (RF = 0.0092 and wRF = 0.0049) and the Fig. 1. The (100) sections of the deformation electron den- ∆ρ sity for crystals (a) I (Nb), (b) II (NbO ), and (c) III highest maxima of the residual electron density ( = 0.15 3 3 (NbO0.107). The isolines are spaced by 0.2 e/Å . 0.6 e/Å ). These maxima are located at a distance of

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STRUCTURE STUDIES OF SOLID SOLUTIONS 241

(a) (a) x =0 x = 0 0.5

0.3 0.8

0.1

–0.1 0.4

–0.3

0 –0.5 (b) (b) 0.5

0.3 0.8

0.1 c / Å z , z –0.1 0.4

–0.3

0 –0.5 (c) (c) 0.5

0.3 0.8

0.1

–0.1 0.4

–0.3

0 –0.5 –0.3 –0.1 0.1 0.3 0.5 0 0.4 0.8 / y, Å y b

Fig. 3. The (100) sections of the difference electron density Fig. 2. The (100) sections of the probability density func- maps for crystals (a) I, (b) II, and (c) III. The isolines are tions for displacements of Nb atoms in crystals (a) I, (b) II, spaced by 0.2 e/Å3. For crystal I, the difference synthesis and (c) III. was simultaneously residual.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 242 ARAKCHEEVA et al.

(a) lent reﬂections (Rint = 0.025) and, consequently, by 0 x = lower ﬁnal R factors (RF = 0.0072 and wRF = 0.0041). The maximum ∆ρ value (0.69 e/Å3) in the difference electron density map is observed in the position 12d: 0.8 (0.5 0.25 0) (Fig. 3b), which corresponds to the tetrahedral cavity in the bcc Nb structure. Based on the fact that the corresponding map for crystal I has no maximum in this position (Fig. 3a), whereas other characteristics of the maps for crystal I are close to those for crystal II, we localized the O atoms in the tetrahedral 0.4 cavities, whose centers are separated from the four nearest Nb atoms by distances of ~1.85 Å. The lowest R factor was obtained for the thermal parameter Biso = 2.2(2) Å2 and the occupancy of the oxygen position q = 0.025(1), which corresponds to the composition 0 NbO0.150(6). This composition agrees satisfactorily with

c (b) / that of the corresponding cathodic product (NbO0.17(1)) z determined by high-temperature oxidation (Table 1). The residual electron density map for crystal II (Fig. 4a) contains no additional signiﬁcant maxima compared to 0.8 the corresponding map for crystal I (Fig. 3a). In spite of the large parameters of anharmonicity of the Nb atoms in crystal II compared to those in crystal I, the amplitudes of the thermal displacements of Nb in crystal II are half as large as those in crystal I (Table 2; 0.4 Figs. 1, 2).

The lowest final R factors (RF = 0.0061 and wRF = 0.0031) and the smallest maxima in the residual electron density map (∆ρ = 0.30 e/Å3) for crystal III correlate with the fact that it has the smallest volume (Vcr = 0 0.0012 mm3) and is characterized by the lowest R factor 0 0.4 0.8 for averaging equivalent reflections (Rint = 0.022). In y/b the difference electron density map for crystal III, the ∆ρ 3 Fig. 4. The (100) sections of the residual difference electron highest peaks ( = 0.80 e/Å ) correspond to the octa- density maps for crystals (a) II and (b) III. The isolines are hedral cavity in the Nb structure, i.e., to the position 6b: spaced by 0.2 e/Å3. (0.5 0.5 0) (Fig. 3c). No high ∆ρ values corresponding to this position are observed in the maps for crystals I 0.65 Å from the Nb atom in the {100} directions and II (Figs. 3a and 3b, respectively). The distances (Fig. 3a). No other significant maxima that could be from the center of the octahedral cavity to the two near- identified as O atoms were revealed in this map. This is est Nb atoms and the four more remote Nb atoms are in good agreement with the fact that crystal I was taken 1.66 Å and 2.34 Å, respectively. The second highest from the sample of the cathodic precipitate with the maximum (∆ρ = 0.4 e/Å3) corresponds to the position minimum oxygen content (NbOx < 0.01), and its unit-cell 12d: (0.5 0.25 0), like that observed in the map for crys- parameter (a = 3.2969 Å) is close to the tabulated value tal II (Fig. 3). The successive localization of the O atoms for oxygen-free Nb (Table 1). This crystal, which is vir- in both these positions led to the minimum R factors for 2 tually devoid of oxygen, is characterized by the maxi- qoct = 0.018(1) (Biso = 2.1(7) Å ) and qtetr = 0.0089(3) mum amplitudes of thermal vibrations of the Nb atom (B = 0.49(8) Å2). Based on the results of the refine- (Table 2) and by the substantially higher probability of iso its thermal displacement compared to those observed in ment, crystal III has the composition NbO0.107(5). The oxygen-containing crystals II and III. residual electron density map for crystal III (Fig. 4b) is very similar to that for crystal II (Fig. 4a) and contains 3 The volume of crystal II (Vcryst = 0.0088 mm ) is no significant maxima. The character of the thermal 30% smaller than that of crystal I. In addition, crystal II is displacements of the Nb atoms in III is identical to that characterized by a better R factor for averaging equiva- in II (Table 2; Figs. 1b, 1c, 2b, 2c).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STRUCTURE STUDIES OF SOLID SOLUTIONS 243

RESULTS AND DISCUSSION of the O and Nb atoms, whereas O atoms are inserted into the already existing structure of Nb upon surface The positions of the O atoms in the structures of oxidation. Hence, the results of our study provide cubic solid solutions of composition NbOx were local- experimental evidence for the earlier assumption [12] ized based on comparison of their difference electron that the positions that are occupied by oxygen atoms density maps with those for the structure of oxygen- depend on the mode of formation of the NbO crystal. free Nb. All maps were constructed using identical sets x We demonstrated that NbOx crystals prepared by elec- of structure factors, which were obtained under the trochemical crystallization and surface oxidation con- same experimental conditions from crystals character- tained oxygen atoms in the tetrahedral and octahedral ized by virtually identical habit and perfection, which cavities, respectively, of the bcc Nb structure. allowed us to reveal essential differences in the crystal structures without resorting to analysis of the details of When comparing the amounts of oxygen dissolved the difference electron density maps associated with the in the bcc Nb structure and the cubic unit-cell parame- conditions of X-ray data collections. The difference ters for crystals II and III (Table 1), it can be concluded that the arrangement of O atoms in the octahedral cav- electron density maps for NbOx differ essentially from the map calculated for Nb in that the former maps have ities of the structure (crystal III) leads to a sharper maximum ∆ρ values corresponding to the centers of the increase in the a unit-cell parameter compared to crys- tetrahedral and octahedral cavities in the bcc Nb struc- tals in which oxygen atoms occupy tetrahedral cavities ture, which were identified as oxygen positions. The (crystal II). The lower oxygen content in crystal III corresponds to the larger a parameter. This fact is attrib- fact that the residual electron density for NbOx is similar to that for Nb confirmed the validity of the determi- utable to the presence of two Nb–O distances (~1.66 Å) nation and the numerical characteristics of the oxygen that are too short in the octahedra along the [100] direc- positions. The positions of the O atoms in the Nb struction. The tendency of Nb atoms to move apart along this ture were revealed for the first time by direct X-ray dif- direction leads to an increase in the unit-cell parameter. fraction methods. As mentioned above, the theoretical As mentioned above, the amplitudes of thermal dis- studies performed earlier provided evidence that oxy- placements of the Nb atoms in crystal I, which is virtu- gen atoms can occupy both octahedral [13] and tetrahe- ally devoid of oxygen, are twice as large as those in dral [14] cavities in the bcc Nb structure. solid solutions of composition NbOx (crystals II and III). This fact is indicative of the substantial influence The results of our study also demonstrated that the of oxygen on the dynamics of the crystal lattice of nio- arrangement of the O atoms depends on the mode of bium metal. On the whole, an increase in the numerical preparation of the cubic solid solution of NbOx. In the parameters of the anharmonicity tensor for NbOx com- case of electrochemical precipitation on the cathode, pared to those for oxygen-free Nb (Table 2) is consis- which was used for the preparation of crystals II, both tent with the statement that the insertion of additional Nb and O atoms present in the melt were involved in the atoms into the initial structure leads to an increase in formation of the crystal structure, and the oxygen the anharmonicity. atoms in the crystal occupied the tetrahedral cavities in the Nb structure. Upon surface oxidation of Nb crystals in a solution of acids, O atoms occupy predominantly CONCLUSIONS the octahedral cavities (the occupancy of the octahedral cavities is twice as large as that of the tetrahedral cavi- It is very interesting that the maximum concentra- ties, see Table 2). Taking into account that O atoms in tion of oxygen in cubic solid solutions of composition electrolytically prepared crystals II occupy only the tet- NbOx, which were prepared by the electrolysis of salt rahedral cavities of the body-centered cubic structure of melts (up to 17 at. %, i.e., up to the composition Nb, we believe that O atoms in crystal III also occupied NbOx = 0.17), is many times higher than all known con- the tetrahedral cavities in the course of formation of the centrations for solid solutions obtained by heating Nb crystal on the cathode, whereas the octahedral cavities in air (0.5Ð1.5 at. % at 700Ð800°C and no higher than were filled with O atoms upon its subsequent surface 7 at. % at 1700°C) [8, 15, 20, 21]. Apparently, the exter- oxidation in the solution of acids. The assumption that nal pressure should increase the solubility of O in Nb, the arrangement of oxygen atoms in the Nb structure and it can be concluded that electrolytic crystallization depends on the mode of preparation of the solid solu- of NbOx solid solutions on the cathode exerts an effect tion has been made in [12] based on the comparison of analogous to that which appears under high external the characteristic features of the preparation of oxygen- pressure. This conclusion is supported by the results of containing niobium compounds by gas-phase reactions our investigation. Thus, oxygen atoms in crystals II with those using thermal surface oxidation of crystals prepared by the electrolytic method are located in the in air according to the mechanism of chemisorption tetrahedral cavities of the bcc Nb structure, whose vol- (the latter has been described in detail [5]). The authors ume is four times smaller than the volume of the octa- of the cited study [12] also believed that the crystal hedral cavities occupied by oxygen upon surface oxida- structure of the solid phase is formed in the course of tion. In addition, the volume of the cubic unit cell of gas-phase reactions with the simultaneous participation crystal II of composition NbOx = 0.15, which was pre-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 244 ARAKCHEEVA et al. pared by the electrolytic method, (36.2 Å3) is 4.3% 6. R. P. Elliott, Constitution of Binary Alloys (McGraw- smaller than the volume of the tetragonally distorted Hill, New York, 1965; Metallurgiya, Moscow, 1970), 3 Vols. 1, 2. unit cell of suboxide Nb6O (37.78 Å ) of virtually the same composition (NbO ) [4] and is 0.3% smaller 7. O. P. Kolchin and I. V. Sumarokowa, At. Energ. 45, 293 0.143 (1978). than the volume of the unit cell of crystal III, which 8. J. D. Fast, Interaction of Metals and Gases (Academic, was prepared by surface oxidation and contained a New York, 1965). smaller amount of oxygen (NbO ). Earlier [22], x = 0.107 9. T. Hurlen, J. Inst. Met. 89, 273 (1961). we drew the conclusion that electrochemical crystalli- 10. V. V. Grinevich, M. S. Model’, A. V. Arakcheeva, et al., zation on the cathode exerts an effect analogous to the Dokl. Akad. Nauk SSSR 319 (2), 389 (1991). rise in the external pressure based on an analysis of the 11. A. V. Arakcheeva, Kristallografiya 37, 589 (1992) [Sov. characteristic features of the tetragonal modification of Phys. Crystallogr. 37, 306 (1992)]. β-Ta. This modification is generated only by electroly- 12. V. I. Kobyakov and V. N. Taranovskaya, Kristallografiya sis and is absent in the known TÐP phase diagrams for Ta. 44, 1017 (1999) [Crystallogr. Rep. 44, 948 (1999)]. The structural defects caused by the replacement of 13. V. K. Grigorovich and E. N. Sheftel’, Dispersion Hard- K atoms by TaÐTa dumbbells were observed in the ening of Refractory Metals (Nauka, Moscow, 1980). cubic structure of KTa O bronze, which was also 1 + z 3 14. B. M. Zykov and A. M. Sabel’nikov, Poverkhnost, prepared by the electrolysis of a salt melt [23]. These No. 10, 61 (1988). defects are indicative of the presence of local centers 15. Binary Alloy Phase Diagrams, Ed. by Th. B. Massalski with a higher density in this perovskite-like structure, (American Society for Metals, Metals Park, 1987, 2nd which can be interpreted as the manifestation of the ed.), Vol. 2. effect analogous to that of external pressure. The syn- 16. V. V. Grinevich, V. A. Reznichenko, G. A. Menyaœlova, thesis of diamond and diamond-like nucleation centers and Yu. I. Bykovskaya, in Electrolytic Dissolution of (nanosized) upon electrolysis of alcoholic solutions Niobium in Molten Salts (Nauka, Moscow, 1975), p. 220. [24–27] also confirms the above assumptions. 17. V. V. Grinevitch, V. A. Reznichenko, M. S. Model, et al., J. Appl. Electrochem. 29, 693 (1999). 18. V. V. Grinevitch, A. V. Arakcheeva, E. G. Polyakov, ACKNOWLEDGMENTS et al., in Proceedings of the International Symposium on Molten Salt XI (The Electrochemical Society, Penning- This study was supported by the Russian Founda- ton, 1998), p. 84. tion for Basic Research. A.V. Arakcheeva acknowl- 19. J. Emsley, The Elements (Clarendon, Oxford, 1991; Mir, edges the financial support of the Herbette Foundation Moscow, 1993). of the University of Lausanne (Switzerland). 20. C. J. Altstetter, Bull. Alloy Phase Diagrams 5 (6), 543 (1984). 21. E. Fromm and E. Gebhardt, Gase und Kohlenstoffin Met- REFERENCES allen (Springer-Verlag, Berlin, 1976; Metallurgiya, 1. N. P. Lyakishev, N. A. Tulin, and Yu. L. Pliner, Doped Moscow, 1980). Alloys and Steels with Niobium (Metallurgiya, Moscow, 22. A. V. Arakcheeva, G. Chapuis, and V. Grinevitch, Acta 1981). Crystallogr., Sect. B: Struct. Sci. B58, 1 (2002). 23. A. V. Arakcheeva, G. Chapuis, V. Grinivitch, and 2. B. Gunter, D. Lieb, C. Richmond, and F. Rufeh, Sum- V. F. Shamray, Acta Crystallogr., Sect. B: Struct. Sci. B7, mary Report on Oxygen Additives (Waltham, Mass., 157 (2001). 1974). 24. Z. Sun, X. Wang, and Y. Sun, Mater. Sci. Eng. B 65, 194 3. I. G. Gverdtsiteli, N. E. Menabde, L. M. Tsakadze, and (1999). V. K. Tskhakaya, in Thermionic Conversion of Heat 25. Y. Namba, J. Vac. Sci. Technol. A 10, 3368 (1992). Energy to Electric Energy (Fiziko-Énergeticheskiœ Inst., 26. H. Wang, M. R. Shen, Z. Y. Ning, et al., Appl. Phys. Lett. Obninsk, 1980), p. 160. 69, 1074 (1996). 4. G. Brauer, H. Muller, and G. Kuhner, J. Less-Common 27. M. C. Tosin, A. C. Peterlevitz, G. L. Surdutovich, and Met. 4 (6), 533 (1962). V. Baranauskas, Appl. Surf. Sci. 144Ð145, 260 (1999). 5. F. Fairbrother, The Chemistry of Niobium and Tantalum (Elsevier, Amsterdam, 1967; Khimiya, Moscow, 1972). Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

STRUCTURE OF COORDINATION COMPOUNDS

Structures of Mixed-Ligand Manganese(II) Compounds [Mn(Heida)(Phen)]2 ◊ 7H2O and [Mn2(Edta)(Phen)] ◊ 4H2O I. N. Polyakova*, V. S. Sergienko*, and A. L. Poznyak** * Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Leninskiœ pr. 31, Moscow, 119991 Russia e-mail: [email protected] ** Institute of Molecular and Atomic Physics, Belarussian Academy of Sciences, pr. F. Skoriny 70, Minsk, 220072 Belarus Received June 5, 2001

Abstract—The crystal structures of [Mn(Heida)(Phen)]2 á 7H2O (I) and [Mn2(Edta)(Phen)] á 4H2O (II) are studied by X-ray diffraction [R1 = 0.0375 (0.0283) and wR2 = 0.0954 (0.0662) for 5449 (3176) observed reﬂec- tions in I (II), respectively]. Structure I contains mononuclear mixed-ligand complexes [Mn(Heida)(Phen)] and 2Ð 2+ [Mn(Heida)(Phen)(H2O)]. In structure II, the [Mn(Edta)] anionic complexes and the [Mn(Phen)(H2O)2] cationic complexes are linked by the bridging carboxyl groups into the tetramers with C2 symmetry. In both compounds, two independent Mn atoms have different coordination numbers (6 and 7). © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION ity of the twofold rotation axis and the center of sym- Continuing our studies of the transition-metal metry, are populated by half. The oxygen atom of the mixed-ligand compounds with aminocarboxylic acids w4 water molecule is disordered over two equally pop- and N-heterocyclic ligands, we determined the crystal ulated positions (A and B) so that the O(4wB) (x, y, z) position is occupied in combination with O(5w) (x, y, z). structures of [Mn(Heida)(Phen)2 á 7H2O (I) and [Mn2(Edta)(Phen)] á 4H2O (II), where H2Heida is hydroxyethyliminodiacetic acid, H4Edta is ethylenedi- RESULTS AND DISCUSSION aminetetraacetic acid, and Phen is 1,10-phenanthroline. Structure of crystals I. Crystals I are built of monomeric mixed-ligand complexes of two types, EXPERIMENTAL [Mn(1)(Heida)(Phen)(H2O)] and [Mn(2)(Heida)(Phen)] Crystals I and II were prepared by slow evaporation (Fig. 1), and molecules of crystallization water. The of aqueous solutions containing the monoligand com- coordination of both Mn atoms includes three O atoms and the N atom of the Heida2Ð tetradentate ligand and pound [Mn(Heida) á 2H2O or Mn3(HEdta)2 á 10H2O] and Phen in equimolar amounts. two N atoms of the Phen molecules. The Mn(1) atom is additionally coordinated by the O(1w) atom of the X-ray diffraction study. The main crystal data and water molecule. Thus, the Mn(1) and Mn(2) atoms have characteristics of the experiment and refinement for I different coordination numbers (7 and 6, respectively). and II are summarized in Table 1. The experimental The Mn(2) polyhedron is a distorted trigonal prism. sets of the I(hkl) intensities for crystals I and II were Triangular bases of the prism, namely, obtained on an EnrafÐNonius CAD4 automated diffrac- λ ω N(2A)N(3A)O(3A) and O(5A)O(1A)N(1A), form an tometer ( MoKα, graphite monochromator, scan angle of 7.6°. In the projection onto the mode). N(2A)N(3A)O(3A) plane, the Mn(2)ÐO(1A) bond coin- Structures I and II were solved by the direct method cides with the Mn(2)ÐN(3A) bond, whereas two other (SHELXS86 [1]). All the H atoms were located from pairs of bonds diverge noticeably: the difference Fourier syntheses and refined isotropically. N(1A)Mn(2)O(3A) and O(5A)Mn(2)N(2A) pseudoan- The non-hydrogen atoms were refined in the anisotro- gles that characterize this divergence are 15° and 50°, pic approximation. For both crystals, the data were cor- respectively. The Mn(1) polyhedron is a similar trigo- rected for absorption according to the azimuthal-scan nal prism with the N(2)N(3)O(3) and O(5)O(1)N(1) technique. The refinement was performed with the bases, but the O(1)O(5)N(2)N(3) face is centered by the SHELXL97 program [2]. O(1w) atom. The angle between the planes of the trian- The atomic coordinates and thermal parameters are gular bases is 5.5°. In the projection onto the listed in Table 2. In structure II, the positions of the w3 N(2)N(3)O(3) plane, the Mn(1)ÐN(1) bond coincides and w5 water molecules, which are located in the vicin- with the Mn(1)ÐO(3) bond, whereas two other pairs of

Table 1. Main crystal data and parameters of data collection and reﬁnement for structures I and II Parameter III

Empirical formula C18H24MnN3O8.5 C22H28Mn2N4O12 M 473.34 650.36 Space group P1 C2/c a, Å 10.276(2) 18.887(4) b, Å 13.675(3) 14.584(3) c, Å 14.989(3) 20.085(4) α, deg 77.61(3) 90 β, deg 82.72(3) 108.45(3) γ, deg 89.91(3) 90 V, Å3 2040.0(7) 5248(1) Z 48 ρ(calcd), g/cm3 1.541 1.646 Crystal size, mm 0.12 × 0.21 × 0.75 0.09 × 0.21 × 0.33 µ Ð1 Mo, mm 0.701 1.033 θ max, deg 27 28 Number of reﬂections: measured 8220 5435

unique (N1) [Rint] 7812 [0.0169] 5281 [0.0256] σ with I > 2 (I), (N2) 5449 3176 R1, wR2 for N1 0.0693, 0.1082 0.0889, 0.0787 R1, wR2 for N2 0.0375, 0.0954 0.0283, 0.0662 GOOF 0.998 0.981 ∆ρ ∆ρ Ð3 min and max, e Å Ð0.681 and 0.538 Ð0.519 and 0.303 bonds diverge to an even larger degree than those in tion to the N + 3O atoms of the Heida ligand, the coor- the Mn(2) polyhedron: the N(2)Mn(1)O(5) and dination of the Mn atom includes the water molecule N(3)Mn(1)O(1) pseudoangles are 38° and 57°, respec- and the carbonyl O atom of the neighboring complex in tively. the trans positions with respect to the oxygen atom of the hydroxyethyl group and the nitrogen atom, respec- The w(1) water molecule “pushes” the O(1), N(2), tively. In the [Mn(Heida)(Phen)] complex of com- and N(3) atoms in the coordination sphere of the Mn(1) pound I, the N(2A) and N(3A) atoms of the Phen mol- atom. The angles involving these atoms change most of ecule are situated trans relative to N(1A) and the car- all in the Mn(1) polyhedron as compared to the corre- boxyl O(1A) atom. This indicates that, when interacting sponding angles in the Mn(2) polyhedron: with Phen in a solution, the initial monoligand complex O(1)MnO(3), 97.51(7)° and 140.89(8)°; O(1)MnO(5), undergoes a transformation. 122.54(8)° and 88.51(8)°; O(1)MnN(2), 155.87(8)° and 120.44(8)°; O(3)MnN(2), 80.56(7)° and 96.16(8)°; In crystal I, ligands of the same type (both Heida N(1)MnN(3), 142.14(8)° and 126.82(8)°; and and Phen) are packed into separate layers, which are N(1)MnN(2), 130.86(8)° and 155.91(8)° for Mn(1) and parallel to the xy plane and alternate in the direction of Mn(2), respectively. The bonds with the Heida2Ð and the z-axis. The water molecules are situated inside the Phen ligands in the seven-coordinate complex of Mn(1) hydrophilic Heida layer and, together with the OH are substantially elongated in comparison with the cor- groups of the hydroxyethyl branches, form an extensive responding bonds in the Mn(2) complex [MnÐO, system of hydrogen bonds (Table 3). The Phen mole- 2.201(2)–2.292(2) and 2.124(2)–2.239(2) Å; Mn–N(1), cules from different complexes [Mn(1) and Mn(2)] are 2.427(2) and 2.314(2) Å; Mn–N(2), 2.303(2) and approximately parallel to each other (the dihedral angle 2.241(2) Å; and Mn–N(3), 2.365(2) and 2.246(2) Å in between their mean planes is 8.6°) and to the xz plane. the Mn(1) and Mn(2) complexes, respectively]. The Phen molecules of the identical complexes form centrosymmetric pairs with interplanar spacings equal The crystals of the initial compound to 3.40 and 3.35 Å. These pairs are arranged in such a Mn(Heida) á 2H2O have a chain structure [3]. In addi- way that, in the projection onto their own plane, each

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STRUCTURES OF MIXED-LIGAND MANGANESE(II) COMPOUNDS 247

Table 2. Coordinates and parameters of thermal vibrations Ueq of the non-hydrogen atoms in structures I and II 2 2 Atom xyzUeq, Å Atom xyzUeq, Å I Mn(1) 0.57101(4) 0.11702(3) 0.75551(2) 0.0271(1) O(4A) 0.9226(2) 0.9153(1) 0.7553(2) 0.0446(5) O(1) 0.4090(2) 0.1099(2) 0.8719(1) 0.0451(5) O(5A) 0.6229(2) 0.5940(2) 0.7609(1) 0.0434(5) O(2) 0.3539(2) 0.1012(2) 1.0214(1) 0.0396(4) N(1A) 0.8058(2) 0.6634(2) 0.8605(1) 0.0315(5) O(3) 0.5638(2) 0.2784(1) 0.6956(1) 0.0342(4) N(2A) 0.7883(2) 0.6316(2) 0.5723(1) 0.0300(5) O(4) 0.5323(2) 0.4322(1) 0.7175(1) 0.0397(4) N(3A) 1.0289(2) 0.6173(2) 0.6292(1) 0.0322(5) O(5) 0.7837(2) 0.0814(2) 0.7788(2) 0.0422(5) C(1A) 0.8472(4) 0.5766(2) 0.9279(2) 0.0406(7) N(1) 0.6358(2) 0.2072(2) 0.8665(1) 0.0285(4) C(2A) 0.8759(3) 0.4851(2) 0.8878(2) 0.0335(6) N(2) 0.6633(2) 0.1183(2) 0.6025(1) 0.0327(5) C(3A) 0.8802(3) 0.7563(2) 0.8561(2) 0.0407(7) N(3) 0.4035(2) 0.1015(2) 0.6705(1) 0.0330(5) C(4A) 0.8901(3) 0.8263(2) 0.7607(2) 0.0338(6) C(1) 0.5735(3) 0.1522(2) 0.9566(2) 0.0327(6) C(5A) 0.6636(3) 0.6805(2) 0.8761(2) 0.0449(7) C(2) 0.4346(3) 0.1188(2) 0.9498(2) 0.0322(6) C(6A) 0.5840(3) 0.5986(3) 0.8547(2) 0.0498(8) C(3) 0.5835(3) 0.3080(2) 0.8446(2) 0.0367(6) C(7A) 0.6705(3) 0.6383(2) 0.5435(2) 0.0372(6) C(4) 0.5588(2) 0.3416(2) 0.7446(2) 0.0284(5) C(8A) 0.6509(3) 0.6475(2) 0.4522(2) 0.0429(7) C(5) 0.7795(3) 0.2109(2) 0.8603(2) 0.0350(6) C(9A) 0.7570(3) 0.6486(2) 0.3878(2) 0.0425(7) C(6) 0.8368(3) 0.1122(2) 0.8514(2) 0.0382(6) C(10A) 0.8838(3) 0.6395(2) 0.4141(2) 0.0345(6) C(7) 0.7880(3) 0.1257(2) 0.5682(2) 0.0412(6) C(11A) 0.9983(3) 0.6348(2) 0.3517(2) 0.0438(7) C(8) 0.8341(4) 0.1392(2) 0.4745(2) 0.0491(8) C(12A) 1.1163(3) 0.6236(2) 0.3804(2) 0.0442(7) C(9) 0.7448(4) 0.1448(2) 0.4135(2) 0.0489(8) C(13A) 1.1330(3) 0.6183(2) 0.4748(2) 0.0361(6) C(10) 0.6109(3) 0.1372(2) 0.4464(2) 0.0405(7) C(14A) 1.2545(3) 0.6072(2) 0.5081(2) 0.0441(7) C(11) 0.5109(4) 0.1413(2) 0.3871(2) 0.0501(8) C(15A) 1.2603(3) 0.6000(2) 0.5995(2) 0.0477(7) C(12) 0.3836(4) 0.1304(2) 0.4208(2) 0.0493(8) C(16A) 1.1463(3) 0.6060(2) 0.6581(2) 0.0422(7) C(13) 0.3421(3) 0.1162(2) 0.5177(2) 0.0402(7) C(17A) 1.0223(3) 0.6229(2) 0.5386(2) 0.0289(5) C(14) 0.2102(3) 0.1049(2) 0.5564(2) 0.0493(8) C(18A) 0.8939(3) 0.6324(2) 0.5083(2) 0.0282(5) C(15) 0.1777(3) 0.0915(3) 0.6488(3) 0.0542(8) O(1w) 0.5608(2) Ð0.0512(2) 0.8062(2) 0.0399(5) C(16) 0.2776(3) 0.0908(2) 0.7043(2) 0.0421(7) O(2w) 0.6975(3) 0.5340(2) 0.1633(2) 0.0691(8) C(17) 0.4372(3) 0.1138(2) 0.5780(2) 0.0303(5) O(3w) 1.0793(2) Ð0.0650(2) Ð0.1095(2) 0.0501(5) C(18) 0.5733(3) 0.1235(2) 0.5426(2) 0.0322(6) O(4w) 1.1130(2) 0.1106(2) Ð0.0456(2) 0.0464(5) Mn(2) 0.83664(4) 0.62937(3) 0.71467(2) 0.0292(1) O(5w) 0.8863(3) 0.6836(2) 0.1376(2) 0.0613(7) O(1A) 0.8791(2) 0.4925(1) 0.8025(1) 0.0452(5) O(6w) 0.6836(3) 0.1547(2) 0.1698(2) 0.0567(6) O(2A) 0.8973(2) 0.4061(2) 0.9430(1) 0.0481(5) O(7w) 0.7789(3) 0.3534(2) 0.1226(2) 0.0727(8) O(3A) 0.8704(2) 0.7885(1) 0.6940(1) 0.0399(4) II Mn(1) 0.08620(2) 0.09652(3) 0.22297(2) 0.0215(1) C(8) 0.1783(2) 0.2668(2) 0.2636(2) 0.0349(7) Mn(2) 0.00082(2) 0.17946(3) 0.02785(2) 0.0243(1) C(9) 0.2311(2) Ð0.0142(2) 0.3097(2) 0.0284(6) O(1) 0.0357(1) 0.0950(1) 0.3188(1) 0.0260(4) C(10) 0.2609(2) 0.0763(2) 0.2933(2) 0.0316(7) O(2) 0.0770(1) 0.1076(1) 0.4354(1) 0.0307(4) C(11) 0.1249(2) 0.2617(3) Ð0.0355(2) 0.053(1) O(3) 0.0769(1) Ð0.0454(1) 0.1799(1) 0.0303(4) C(12) 0.1691(2) 0.3275(3) Ð0.0516(2) 0.068(1) O(4) 0.0868(1) Ð0.1952(1) 0.2013(1) 0.0385(5) C(13) 0.1682(2) 0.4144(3) Ð0.0282(2) 0.059(1) O(5) 0.0939(1) 0.1186(1) 0.1098(1) 0.0269(4) C(14) 0.1234(2) 0.4361(2) 0.0136(2) 0.0407(8) O(6) 0.1548(1) 0.0480(2) 0.0452(1) 0.0422(5) C(15) 0.1168(2) 0.5260(2) 0.0398(2) 0.050(1) O(7) 0.1193(1) 0.2292(1) 0.2670(1) 0.0405(5) C(16) 0.0759(2) 0.5412(2) 0.0819(2) 0.048(1) O(8) 0.2023(1) 0.3431(2) 0.2885(1) 0.0529(6) C(17) 0.0358(2) 0.4688(2) 0.1023(2) 0.0383(8) N(1) 0.1545(1) Ð0.0036(1) 0.3138(1) 0.0233(5) C(18) Ð0.0077(2) 0.4822(2) 0.1467(2) 0.0506(9) N(2) 0.2107(1) 0.1151(1) 0.2267(1) 0.0252(5) C(19) Ð0.0470(2) 0.4099(3) 0.1604(2) 0.0525(9) N(3) 0.0820(1) 0.2780(2) 0.0041(1) 0.0353(6) C(20) Ð0.0429(2) 0.3248(2) 0.1300(2) 0.0424(8) N(4) Ð0.0021(1) 0.3095(2) 0.0886(1) 0.0317(5) C(21) 0.0380(2) 0.3806(2) 0.0751(1) 0.0313(7) C(1) 0.1552(2) 0.0339(2) 0.3817(1) 0.0255(6) C(22) 0.0820(2) 0.3641(2) 0.0299(1) 0.0307(6) C(2) 0.0830(1) 0.0826(2) 0.3781(1) 0.0225(5) O(1w) 0.0156(1) 0.0836(2) Ð0.0478(1) 0.0407(6) C(3) 0.1157(2) Ð0.0923(2) 0.2991(1) 0.0256(6) O(2w) Ð0.0912(2) 0.2296(2) Ð0.0638(1) 0.0403(6) C(4) 0.0917(2) Ð0.1139(2) 0.2204(1) 0.0267(6) O(3w)* 0.009(2) 0.6743(5) 0.261(2) 0.134(8) C(5) 0.2192(2) 0.0676(2) 0.1658(1) 0.0293(6) O(4wA)* 0.7267(4) 0.3667(5) 0.0681(4) 0.049(2) C(6) 0.1508(2) 0.0788(2) 0.1012(1) 0.0264(6) O(4wB)* 0.7594(4) 0.3913(5) 0.0702(5) 0.051(2) C(7) 0.2224(2) 0.2149(2) 0.2242(2) 0.0339(7) O(5w)* 0.7220(3) 0.2115(4) 0.0383(3) 0.057(1) * Site occupancy is 0.5.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 248 POLYAKOVA et al.

(a) O(2)

C(15) O(1) C(2) C(16) C(14)

C(1) N(3) O(1W) (b) C(13) Mn(1) N(1) C(3) C(6 ) C(17) A C(12) C(18) C(4) O(3) O(4) O(5A) C(5A) C(11) O(5) C(5) O(2A) C(1A) C(10) N(2) C(2A) C(6) O(1A) N(1A) C(7A) C(7) C(8A) C(3A) C(9) N(2A) Mn(2) C(8) C(9A) O(3A) C(4A) C(18A) O(4A) C(10A) N(3A) C(17A) C(11A) C(16A) C(13A) C(12A)

C(15A) C(14A)

Fig. 1. Structure of (a) [Mn(Heida)(Phen)(H2O)] and (b) [Mn(Heida)(Phen)] complexes in I. The H atoms are omitted.

C(10) O(8) C(7) C(5) C(9) C(13) N(2) O(6) C(8) N(1) C(12) C(14) C(1) C(6) C(15) O(7) C(11) C(16) O(5) C(3) C(2) N(3) O(2) C(22) O(3) C(21) C(17) O(4) Mn(1) O(1W) C(18) O(1)

N(4) Mn(2) C(20) C(19)

O(2W)

Fig. 2. Structure of the {[Mn(Edta)][Mn(Phen)(H2O)2]}2 tetramer in II. The H atoms are omitted.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STRUCTURES OF MIXED-LIGAND MANGANESE(II) COMPOUNDS 249

Table 3. Characteristics of hydrogen bonds in structures I and II

Symmetry transformation Distance, Å XÐH⋅⋅⋅Y XÐH⋅⋅⋅Y for the Y atom X⋅⋅⋅Y H⋅⋅⋅Y angle, deg I O(1w)ÐH(1w1)⋅⋅⋅O(6w) 1 Ð x, Ðy, 1 Ð z 2.829(4) 2.02(4) 174(4) O(1w)ÐH(2w1)⋅⋅⋅O(2) 1 Ð x, Ðy, 2 Ð z 2.779(3) 2.02(4) 174(4) O(2w)ÐH(1w2)⋅⋅⋅O(4) 1 Ð x, 1 Ð y, 1 Ð z 2.875(4) 2.12(4) 158(3) O(2w)ÐH(2w2)⋅⋅⋅O(5w) x, y, z 2.754(5) 1.94(4) 174(4) O(3w)ÐH(1w3)⋅⋅⋅O(4A) x, y Ð 1, z Ð 1 2.799(3) 1.97(4) 167(4) O(3w)ÐH(2w3)⋅⋅⋅O(4w) 2 Ð x, y, z 2.813(4) 1.90(6) 178(6) O(4w)ÐH(1w4)⋅⋅⋅O(2) 1 + x, y, z Ð 1 2.780(3) 2.08(4) 175(4) O(4w)ÐH(2w4)⋅⋅⋅O(3w) x, y, z 2.806(3) 1.70(7) 169(5) O(5w)ÐH(1w5)⋅⋅⋅O(2A) 2 Ð x, 1 Ð y, 1 Ð z 2.803(4) 2.01(4) 164(6) O(5w)ÐH(2w5)⋅⋅⋅O(4w) 2 Ð x, 1 Ð y, Ðz 2.857(4) 2.07(5) 173(5) O(6w)ÐH(1w6)⋅⋅⋅O(7w) x, y, z 2.802(4) 1.94(4) 168(4) O(6w)ÐH(2w6)⋅⋅⋅O(3w) 2 Ð x, Ð y, Ðz 2.857(4) 2.14(5) 161(5) O(7w)ÐH(1w7)⋅⋅⋅O(2A) x, y, z Ð 1 2.755(4) 1.88(5) 175(5) O(7w)ÐH(2w7)⋅⋅⋅O(2w) x, y, z 2.772(4) 1.88(5) 177(4) O(5)ÐH(5O)⋅⋅⋅O(4A) x, y Ð 1, z 2.743(3) 1.99(4) 170(4) O(5A)ÐH(5OA)⋅⋅⋅O(4) x, y, z 2.639(3) 1.79(3) 170(3) II O(1w)ÐH(1w1)⋅⋅⋅O(3) Ðx, Ðy, Ðz 2.732(3) 1.85(4) 172(4) O(1w)ÐH(2w1)⋅⋅⋅O(2) x, Ðy, z Ð 0.5 3.079(3) 2.43(4) 139(4) O(1w)ÐH(2w1)⋅⋅⋅O(6) x, y, z 2.750(3) 2.14(4) 132(4) O(2w)ÐH(1w2)⋅⋅⋅O(4) Ðx, Ðy, Ðz 2.834(3) 2.12(4) 174(4) O(2w)ÐH(2w2)⋅⋅⋅O(5w) 0.5 Ð x, 0.5 Ð y, Ðz 2.812(7) 2.18(5) 154(4) O(2w)ÐH(2w2)⋅⋅⋅O(4wA) 0.5 Ð x, 0.5 Ð y, Ðz 2.897(8) 2.33(5) 142(5) O(3w)ÐH(1w3)⋅⋅⋅O(4) x, 1 + y, z 2.89(3) 2.23(8) 149(8) O(3w)ÐH(1w3)⋅⋅⋅O(4) Ðx, 1 + y, 0.5 Ð z 2.89(3) 2.23(8) 149(8) O(4wA)ÐH(1w4)⋅⋅⋅O(8) 1 Ð x, y, 0.5 Ð z 2.787(8) 1.93(4) 163(4) O(4wA)ÐH(2w4)⋅⋅⋅O(6) 0.5 + x, 0.5 + y, z 2.942(8) 2.01(5) 172(4) O(4wB)ÐH(1w4)⋅⋅⋅O(8) 1 Ð x, y, 0.5 Ð z 2.788(9) 1.93(4) 163(4) O(4wB)ÐH(2w4)⋅⋅⋅O(6) 1 Ð x, y, 0.5 Ð z 2.958(8) 2.01(5) 147(4) O(5w)ÐH(1w5)⋅⋅⋅O(4wA) 1.5 Ð x, 0.5 Ð y, Ðz 2.85(1) 1.98(7) 159(6) O(5w)ÐH(2w5)⋅⋅⋅O(4wB) x, y, z 2.739(9) 1.89(9) 154(7) C(18)ÐH(18)⋅⋅⋅O(3w)Ðx, y, 0.5 Ð z 3.37(2) 2.36(4) 170(3) C(13)ÐH(13)⋅⋅⋅O(4wB) 1 Ð x, 1 Ð y, Ðz 3.368(9) 2.47(4) 164(3) C(20)ÐH(20)⋅⋅⋅O(7) Ðx, y, 0.5 Ð z 3.193(4) 2.42(3) 141(2) C(15)ÐH(15)⋅⋅⋅O(5w) x Ð 0.5, 0.5 + y, z 3.363(7) 2.47(3) 157(3) C(16)ÐH(16)⋅⋅⋅O(2w)Ðx, 1 Ð y, Ðz 3.385(4) 2.65(3) 136(2)

molecule overlaps with two other Phen molecules, the C2 symmetry (Fig. 2) and molecules of crystalliza- which results in the formation of the layer. The shortest tion water. The Mn(1) atom is coordinated by the N(1), intermolecular CáááC distances in the layer are 3.292Ð N(2), O(1), O(3), O(5), and O(7) atoms of the Edta4Ð 3.536 Å. ligand and the O(1)' atom of the neighboring complex Structure of crystals II. Structure II is built of the that is related to the reference one by the twofold rota- {[Mn(1)(Edta)][Mn(2)(Phen)(H2O)2]}2 associates with tion axis. Thus, two anionic complexes are linked by

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 250 POLYAKOVA et al. the O(1) and O(1)' bridging atoms into the dimer. The is larger than that in III [2.386(9) and 2.430(9); Mn(1)ÐO(1)ÐMn(1)'ÐO(1)' four-membered ring is 2.16(2)–2.25(2) Å, respectively]. These differences are almost planar: the angle of folding along the O(1)Ð at least in part attributed to the difference in the func- O(1)' segment is 1.4°. The Mn(1)ÐO(1)ÐMn(1)' bridge tions of the carboxyl groups in the two structures. has an asymmetric structure [2.403(2) and 2.187(2) Å]. The O–HáááO hydrogen bonds involving the w1, w2, The Mn(2) atom is coordinated by the N(3) and N(4) and w4 water molecules link the tetramers into the atoms of the Phen molecule, the O(1w) and O(2w) three-dimensional framework (Table 3). The Phen mol- atoms of water molecules, and the O(5) and O(2)' atoms ecules contribute to the stabilization of the structure of the Edta ligands from the two moieties of the dimer. through the formation of the C–HáááO hydrogen bonds. The coordination numbers of the Mn(1) and Mn(2) are In addition, the stacking interactions link the Phen mol- 7 and 6, respectively. ecules into the centrosymmetric pairs with the interpla- Note that, in the crystals of the initial compound nar spacing equal to 3.32 Å and the shortest intermolec- Mn3(HEdta)2 á 10H2O, the Mn atoms also form com- ular CáááC distances equal to 3.355–3.478 Å. plexes of two types, namely, the anionic complexes Ð Thus, the addition of phenanthroline to aqueous [Mn(HEdta)(H2O)] and the cationic complexes 2+ 2+ solutions of the monoligand Mn complexes with [Mn(H2O)4] , which are linked by the bridging car- Heida2Ð and HEdta3Ð results, in the former case, in the boxyl groups into the centrosymmetric trinuclear asso- formation of the mixed-ligand [Mn(Heida)(Phen)] and ciates [4, 5]. [Mn(Heida)(Phen)(H2O)] complexes in which each A similar division of the Mn2+ ions into the anionic metal atom is chelated by both ligands and, in the latter and cationic complexes with coordination numbers 7 case, in the formation of the separate chelate complexes and 6 is observed in the Edta á 2Ð 2+ [Mn(H2O)4][Mn( )(H2O)] [Mn(Edta)] and [Mn(Phen)(H2O)2] which are linked 4H2O compound (III) [6]. In the anionic moiety, in con- by the bridging carboxyl groups into the tetramers. trast to II, the seventh vertex in the Mn environment is occupied by the water molecule. In the cationic moiety, the Mn octahedron consists of four O atoms of water ACKNOWLEDGMENTS molecules and two O atoms of the carboxyl groups of two neighboring anionic complexes. Thus, the cationic We acknowledge the support of the Russian Foun- and anionic complexes alternate to form the chains. dation for Basic Research in the payment of the license for using the Cambridge Structural Database, project In the series of Mn2+ ethylenediaminetetraacetates no. 99-07-90133. with different cations, the structure of the complex remains the same: the Mn atom is coordinated by the hexadentate Edta4Ð ligand and the water molecule. REFERENCES + + + Some of these compounds (Cat = Li [7], Na , Rb [6], 1. G. M. Sheldrick, Acta Crystallogr., Sect. A: Found. 3+ and Nd [8]), like III, have polymeric structures, and Crystallogr. A46 (6), 467 (1990). + 2+ others (Cat = NH4 and Mg [6]) contain isolated 2. G. M. Sheldrick, SHELXL97: Program for the Refine- anionic complexes. The tetranuclear structure found in ment of Crystal Structures (Univ. of Göttingen, Göttin- compound II has not been observed earlier among the gen, 1997). Mn2+ complexes with Edta4Ð. 3. N. N. Anan’eva, T. N. Polynova, and M. A. Poraœ- The Mn(1) polyhedron in II is a distorted mono- Koshits, Koord. Khim. 7 (10), 1556 (1981). capped trigonal prism with the N(1)O(1)O(3) and 4. S. Richards, B. Pedersen, J. V. Silverton, and J. L. Hoard, N(2)O(5)O(7) triangular bases and the O(1)' atom cen- Inorg. Chem. 3 (1), 27 (1964). tering the O(1)O(3)O(5)O(7) face. The bases of the 5. I. N. Polyakova, A. L. Poznyak, V. S. Sergienko, and prism are almost parallel (the dihedral angle is 2.5°) but L. V. Stopolyanskaya, Kristallografiya 46 (1), 47 (2001) are twisted by an angle of ~20°. In structure III, the [Crystallogr. Rep. 46, 40 (2001)]. seven-vertex polyhedron of the Mn atom is also 6. X. Solans, S. Gali, M. Font-Altaba, et al., Afinidad 45, described as a monocapped trigonal prism with the N 243 (1988). 4Ð and O atoms of the Edta ligand at the vertices of the 7. N. N. Anan’eva, T. N. Polynova, and M. A. Poraœ- bases and the H2O molecule in the “cap.” Koshits, Zh. Strukt. Khim. 15 (2), 261 (1974). Although the seven-coordinate manganese com- 8. Tao Yi, Song Gao, and Biaoguo Li, Polyhedron 17, 2243 plexes in II and III are similar in structure, their MnÐ (1998). Lig bond lengths differ significantly. In II, the MnÐN bonds [2.345(2) and 2.374(2) Å] are shorter and the spread of the MnÐO bond lengths [2.137(2)Ð2.403(2) Å] Translated by I. Polyakova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 251Ð261. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 286Ð296. Original Russian Text Copyright © 2002 by Kuz’mina, I. Perepichka, D. Perepichka, Howard, Bryce.

STRUCTURE OF ORGANIC COMPOUNDS

Supramolecular Architecture of Two Charge-Transfer Complexes Based on 2,7-(X,X)-4,5-Dinitro-9-Dicyanomethylenefluorenes (X = NO2 or CN) and Tetrathiafulvalene L. G. Kuz’mina*, I. F. Perepichka**, D. F. Perepichka**, J. A. K. Howard***, and M. R. Bryce*** * Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Leninskiœ pr. 31, Moscow, 119991 Russia ** Litvinenko Institute of Physicoorganic and Coal Chemistry, National Academy of Sciences of Ukraine, ul. R. Lyuksemburg, Donetsk, 84114 Ukraine *** Chemistry Department, Durham University, Durham DH1 3LE, England e-mail: [email protected] Received March 29, 2001

Abstract—The crystal packings of two charge-transfer complexes based on tetrathiafulvalene and substituted fluorenes—2,4,5,7-tetranitro-9-dicyanomethylenefluorene (in complex I) or 2,7-dicyano-4,5-dinitro-9-dicyanomethylenefluorene (in complex II)—are analyzed. Crystals of complex II involve a third component, namely, C6H5Cl solvate molecules. Crystals of both complexes are characterized by the formation of stacks composed of alternating donor and acceptor molecules and sheets in which the molecules are linked through different-type weak interactions. In structure II, chlorobenzene molecules occupy cavities that are formed in stacks in the vicinity of the tetrathiafulvalene molecules due to the larger difference in size of the donor and acceptor molecules in complex II as compared to that in complex I. The chlorobenzene molecules provide a close packing. These molecules are involved in the system of weak interactions to form the CláááN and C–HáááN secondary bonds with the CN groups of the acceptor molecules in the sheets. © 2002 MAIK “Nauka/Interpe- riodica”.

INTRODUCTION tetrathiafulvalene (TTF) with 7,7,8,8-tetracyanoquin- Molecular organic conductors have been studied odimethane (TCNQ) [2], a great number of donors— extensively over the last two decades [1]. After the dis- tetrathiafulvalene derivatives—have been synthesized covery of the ﬁrst stable charge-transfer complex with [1Ð4]. The chemical diagrams of the corresponding a metallic conductivity, namely, the complex of compounds are shown in Scheme I.

O O NC CN CN CN S S S S N N X S S O S S O TTF BEDO–TTF Y N NC CN N NO2 R NC NC TCNQ DCNQI 2,5-X,Y-DCNQI DTF: R = H, Z = NO2 Z Z DTeF: R = Z = NO2 DDDF: R = NO2, Z = CN

NC CN Scheme I.

However, the structure of acceptors has not been works dealing with charge-transfer complexes and rad- adequately investigated. There had been only a few ical ion salts that contained acceptors other than TCNQ

252 KUZ’MINA et al. molecules [5] until the discovery of the new type of ties and photoconductivity [17, 19Ð21]. Recently, Hori- acceptors, namely, N,N'-dicyano-1,4-quinodiimines uchi et al. [22] revealed a metallic conductivity in (DCNQI) [6Ð9]. Radical ion salts 2,5-X,Y-DCNQI with charge-transfer complexes of bis(ethylene- Li, Na, K, Tl, and Ag possess the properties of metallic dioxy)tetrathiafulvalene (BEDO-TTF) with fluorene semiconductors, whereas the radical ion salts with Cu electron acceptors such as 2,4,7-trinitro-9-dicyanome- have an extremely high metallic (three-dimensional) thylenefluorene (DTF) and 2,4,5,7-tetranitro-9-dicya- conductivity [10]. nomethylenefluorene (DTeF): σ = 65 S cmÐ1 at 298 K The use of substituted fluorenes with different sub- and σ = 390 S cmÐ1 at 8 K for BEDO-TTF : DTF = 2 : 1 stituents in the 2- and 7-positions (and in the 4- and 5- and σ = 18 S cmÐ1 at 298 K and σ = 32 S cmÐ1 at 94 K positions) as acceptors in charge-transfer complexes for BEDO-TTF : DTeF = 2 : 1 (pressed pellets). makes it possible to change substantially the size and In this work, we investigated the crystal structures of the shape of acceptor molecules, thus affecting the two charge-transfer complexes based on substituted flu- packing of the donor and acceptor molecules in crys- orenes and tetrathiafulvalene. Compound I is the tals. The electrical conductivity of these charge-transfer tetrathiafulvalene complex with 2,4,5,7-tetranitro-9- complexes can vary by six orders of magnitude [7Ð10]. dicyanomethylenefluorene (TTF : DTeF = 1 : 1). Com- Undoubtedly, the electrical conductivity should be gov- pound II is the tetrathiafulvalene complex with 2,7-di- erned by the specific features of the supramolecular cyano-4,5-dinitro-9-dicyanomethylenefluorene (DDDF) structure (crystal packing) of charge-transfer com- and chlorobenzene (Sol) C6H5Cl (TTF : DDDF : Sol = plexes. The elucidation of the regularities in the archi- 1 : 1 : 1). These charge-transfer complexes differ in tecture of their crystals is important for the modeling of degrees of misfit in the shape and size of the donor (D) the structure of charge-transfer complexes with the and acceptor (A) molecules. Moreover, different sub- highest conductivity. Recent investigations into the stituents (NO2 and CN groups) in the 2- and 7-positions supramolecular architecture revealed the interrelation of the acceptor molecules impose different geometric between the structure of crystals and their properties constraints on the formation of intermolecular second- [11Ð16]. ary bonds that are responsible for the specific features In the present work, we analyzed how the replace- of the molecular packing motif in crystals. At the same ment of the substituent in the electron-acceptor compo- time, the DTeF and DDDF fluorene acceptors are nent of the charge-transfer complex affects its crystal closely similar in electron-acceptor properties (the packing by using the example of two fluorene charge- electron affinities are 2.71 and 2.77 eV, respectively) transfer complexes with the same tetrathiafulvalene [23]. Therefore, the differences in crystal packings of molecule as a donor component. The properties of these compounds I and II can be determined only by the and related charge-transfer complexes either have aforementioned factors (different sizes and geometric already been understood or are being investigated constraints on the formation of intermolecular second- intensively. In particular, it was established that the ary bonds). electronic properties of fluorene acceptors are compa- The synthesis, properties, and X-ray molecular rable to those of TCNQ and DCNQI derivatives. The structures of these charge-transfer complexes were fluorene acceptors readily form the charge-transfer described in our earlier work [23]. The donor and complexes with aromatic electron donors [17, 18]. acceptor components of the studied complexes with the These complexes often possess semiconductor proper- atomic numbering are shown in Scheme II.

N(2) N(1) N(1) N(2) 16 15 15 16 14 14 O(1) O(8) 9 9 8 1 8 1 S(4) S(1) 13 10 N(6) 13 10 N(3) 6' 2' N(6) 7 2 N(3) 18 7 2 17 12 11 12 11 4' 1' O(7) 6 3 O(2) 6 3 5' 3' 5 4 5 4 S(3) S(2) N(5) O(5) N(5) O(3) O(6) N(4) O(4) N(4) O(4) O(3) O(1) O(2) DTeF DDDF TTF Scheme II.

STACKING MOTIFS IN STRUCTURES I AND II metry and, hence, are disordered over two positions with the same occupancy. These molecules occupy the Two crystallographically independent solvent mole- centers of symmetry of two different systems (i and j) cules in structure II are located at the centers of sym- and exhibit different types of disordering. The type of

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

SUPRAMOLECULAR ARCHITECTURE OF TWO CHARGE-TRANSFER COMPLEXES 253

a 0 b

Fig. 1. Stacking structure of charge-transfer complex I. disordering of the solvent molecules is depicted in arranged at a small angle with respect to the a axis in Scheme III. the crystal. However, the structural function of these molecules is not confined only to ensuring a close crystal packing. Cl(1) Cl(1) Cl(2) Cl(2) In both structures, the mean planes of the donor and Scheme III. acceptor molecules are not strictly parallel to each other. The table presents the short intermolecular dis- The molecule at the left is disordered with respect to tances in the stacking triad with the tetrathiafulvalene the symmetry center of the system i. This center of molecule sandwiched between two acceptor molecules symmetry coincides with the center of the benzene (“above” and “below”). The mutual orientations of ring. The centers of the benzene rings of two disordered components of the molecule at the right are displaced molecules in these triads are shown in Figs. 3 and 4 in from the symmetry center of the system j in opposite projections onto the mean plane of one of the fluorene directions. fragments. It can be seen from these figures that the stacking structures of complexes I and II differ qualita- Crystals I and II are characterized by the same tively. The acceptor molecules with the tetrathiaful- stacking motif of alternating donor and acceptor mole- valene molecule in between are displaced relative to cules (Figs. 1, 2), which is typical of charge-transfer each other in their own planes in structure II to a greater complexes. In both crystals, the stacks are aligned extent than in structure I. In structures I and II, the cen- along the shortest unit-cell dimension. The nearest- neighbor like molecules inside each stack are related by tral tetrathiafulvalene molecules are differently located with respect to the fluorene fragments. In structure I, the translation. The replacement of two NO2 groups in the 2- and 7-positions of fluorene (in molecule I) by the the double bond between heterocycles is actually pro- CN groups (molecule II) leads to an increase in the lin- jected outside the tricyclic system. In structure II, the ear size of the acceptor molecule. Therefore, in order to double bond predominantly lies between the fluorene provide a close crystal packing, an additional structural tricycles of the upper and lower molecules, even though block, namely, the solvent molecule, should be intro- the chlorobenzene molecule is also located between the duced into the crystal lattice of compound II. The chlo- two acceptor molecules. This means that the stacking robenzene solvate molecules that belong to the symme- interaction between the donor and acceptor molecules try center system i fill part of the free intrastack space only slightly depends, at least within certain limits, on and partially occupy cavities between the stacks (Fig. 2). their mutual displacements and rotations in parallel The solvate molecules belonging to the symmetry cen- planes. The data presented in the table also indicate dif- ter system j occupy the interstack space and are ferences in mutual packings of molecules in the stack:

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254 KUZ’MINA et al.

0 c

Fig. 2. Stacking structure of charge-transfer complex II and location of solvate molecules of both types. the short intrastack intermolecular contacts in struc- TWO-DIMENSIONAL MOTIFS tures I and II are different. IN STRUCTURES I AND II Compounds I and II have a common feature in their The crystal structures of both compounds, apart stacking structure—the intermolecular contacts of the from the stacks, involve infinite sheets of molecules tetrathiafulvalene molecule with one acceptor molecule linked through weak interactions. These sheets in struc- are generally shorter than those with the other acceptor tures I and II differ in appearance and geometry. molecule. This implies that the pair donorÐacceptor Figure 5 shows the puckered sheet of the acceptor π−π interactions contribute substantially to the stacking and donor molecules linked through different-type sec- motif of crystal packing. ondary interactions in crystal I. Each donor molecule forms five short contacts containing three sulfur atoms The described (qualitative and quantitative) differ- and two CÐH fragments involved in the hydrogen ences in the stacking motifs of structures I and II can bonds with the CN and NO2 groups of the neighboring be associated with the formation of different systems of acceptor molecules. Each acceptor molecule is intermolecular secondary bonds in the crystals. These involved in five weak interactions with the neighboring bonds are accomplished through different electronÐ donor molecules through the oxygen atoms of the NO2 electron interactions (n–σ*, n–π*, and p–π*) with the groups and the CN group. The contacts S(1)áááO(3) participation of functional groups of the acceptor and (3.10 Å), S(2)···O(8) (3.20 Å), and S(4)···O(5) (3.24 Å) donor molecules. These interactions can change the are somewhat shorter than or comparable to the sum of the van der Waals radii. The O(3) and O(8) oxygen ratio between the highest occupied and lowest unoccu- atoms are virtually aligned with one of the SÐC bonds pied molecular orbital levels in the donor and acceptor of the relevant sulfur atom: the O(3)áááS(1)ÐC(1') angle molecules and, hence, can influence the ability of these is equal to 162.8° and the O(8)áááS(2)ÐC(1') angle is molecules to transfer electrons; i.e., these interactions 162.2°. The observed geometry of the short contacts is can affect the properties of the charge-transfer com- typical [24, 25] and corresponds to the n–σ* interaction plex. In this respect, there is a need to examine the sys- with the participation of the sp2 orbital of the lone elec- tems of directional secondary bonds and the structural tron pair of the oxygen atom in the NO2 group and the units formed in the crystals as a result of these interac- σ* orbital of the SÐC bond. This geometry is in agree- tions. ment with the n–π* interaction involving the p orbital

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Short intrastack contacts (Å) between the tetrathiafulvalene molecule and two substituted fluorene molecules located above (molecule A) and below (molecule B) in structures I and II III TTFÐDTeF(A) TTFÐDTeF(B) TTFÐDDDF(A) TTFÐDDDF(B) S(1)ÐC(8) 3.32 S(1)ÐC(9) 3.63 S(1)ÐC(4) 3.61 S(1)ÐC(11) 3.37 S(2)ÐC(11) 3.22 S(1)ÐC(13) 3.60 S(2)ÐC(2) 3.47 S(1)ÐC(12) 3.42 S(2)ÐC(12) 3.39 S(2)ÐC(10) 3.67 S(2)ÐC(17) 3.55 S(2)ÐC(10) 3.45 S(2)ÐO(4) 3.12 S(2)ÐC(11) 3.66 S(3)ÐC(10) 3.86 S(2)ÐC(1) 3.40 S(4)ÐC(14) 3.20 S(3)ÐC(1) 3.64 S(3)ÐC(1) 3.87 S(2)ÐC(9) 3.73 S(4)ÐC(16) 3.42 S(3)ÐO(1) 3.44 S(4)ÐC(11) 3.72 S(3)ÐC(14) 3.22 S(4)ÐC(15) 3.36 S(4)ÐC(15) 3.64 S(4)ÐC(12) 3.46 S(3)ÐC(15) 3.25 C(1')ÐC(9) 3.29 S(4)ÐN(1) 3.78 S(4)ÐC(13) 3.65 S(4)ÐC(8) 3.25 C(1')ÐC(13) 3.31 C(1')ÐC(9) 3.66 C(1')ÐC(2) 3.75 S(4)ÐC(13) 3.60 C(1')ÐC(10) 3.60 C(1')ÐC(10) 3.63 C(1')ÐC(3) 3.74 C(1')ÐC(9) 3.26 C(2')ÐC(7) 3.52 C(2')ÐC(13) 3.59 C(1')ÐC(4) 3.79 C(1')ÐC(10) 3.38 C(2')ÐC(6) 3.67 C(2')ÐC(12) 3.54 C(1')ÐC(11) 3.84 C(1')ÐC(13) 3.48 C(3')ÐC(5) 3.51 C(2')ÐC(13) 3.59 C(4')ÐC(10) 3.55 C(2')ÐC(4) 3.45 C(3')ÐC(12) 3.45 C(3')ÐC(11) 3.76 C(4')ÐC(1) 3.82 C(2')ÐC(11) 3.63 C(4')ÐC(9) 3.27 C(3')ÐC(12) 3.52 C(4')ÐC(11) 3.72 C(3')ÐC(1) 3.68 C(4')ÐC(14) 3.42 C(6')ÐN(1) 3.50 C(5)ÐC(14) 3.64 C(3')ÐC(2) 3.63 C(6')ÐN(1) 3.38 C(6')ÐC(9) 3.69 C(4')ÐC(13) 3.39 C(6')ÐC(13) 3.62 C(4')ÐC(9) 3.29 C(4')ÐC(14) 3.48 C(5')ÐC(15) 3.39 of the lone electron pair of the sulfur atom and the π* sponds to a very weak interaction with a geometry such orbital of the NO2 group. The O(5)áááS(4)ÐC(4') and that the OáááO line appears to be nearly perpendicular to O(5)áááS(4)ÐC(6') angles are equal to 123.0° and 137.8°, the planes of both NO2 groups (with angles of 85.2° and respectively. However, it is evident that one lone elec- 85.6°). This geometry is most consistent with the π–π* tron pair of the O(5) atom is oriented toward the line interactions between these groups. aligned with the S(4)ÐC(6') bond. No other speciﬁc interactions, except for the afore- π π One further weak interaction with the participation mentioned – * interactions, participate in bonding the of the donor and acceptor molecules is the CÐHáááNC puckered sheets. The stacking of these sheets results in hydrogen bond. The N···H distance (2.47 Å) and the the formation of the parallel alternating stacks angles H(3')áááN(2)ÐC(16) (150.7°) and N(2)áááH(3')Ð …ADADAD… . C(3') (163.5°) have standard values for hydrogen bonds The system of secondary bonds in structure II dif- of this type [26]. It is quite probable that there exists fers essentially from that in structure I. Figure 6 depicts one more hydrogen bond, namely, the C(2')Ð the sheet in structure II. The tracery ribbons formed by H(2')áááO(3) hydrogen bond with the H(2')áááO(3) dis- conjugate centrosymmetric macrocycles involving two tance equal to 2.53 Å. However, the short contact pairs of unlike molecules can be distinguished in the between these atoms can be governed by the geometric sheet of the crystal. It can be seen that macrocycles of constraints on the system of weak interactions. two types alternate in the ribbon. A macrocycle of the ﬁrst type is formed through secondary bonds and has a The weak interactions described above are responsi- large cavity that contains the j-type solvent molecule. ble for the formation of zigzag ribbons in the crystal. The minimum size of this cavity (i.e., the distance The ribbons are linked together into puckered sheets between the acceptor hydrogen atoms oriented toward due to the OáááO interactions between the O(1) and O(7) the interior of the macrocycle) is equal to 6.75 Å. The oxygen atoms of the NO2 groups in the adjacent rib- cavity is large enough to house the solvent molecule bons. The O···O distance (2.87 Å) most likely corre- (see Fig. 6). The macrocycle is formed by secondary

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 256 KUZ’MINA et al.

N(1B) C(6'A) N(2B) N(2A) C(15B) N(1A) S(4A) C(16A) C(16B) C(5'A) C(15A)

C(14B) C(14A) C(4'A)

S(1A) C(9B) S(3A) O(1A) C(9A) O(8B) O(8A) C(8B) C(8A) C(10A) C(1A) O(1B) C(1'A) N(3A) N(6A) C(13A) C(1B) C(2A) C(13 ) B C(10B) N(6B) C(2'A) C(7A) C(7B) C(12A) C(11A) C(2B) N(3B) O(2A) C(12B) O(7A) S(2A) C(4A) O(7B) C(3A) C(6A) O(2B) C(3'A)C(11B) C(5A) C(3B) C(6B) C(4B) C(5B)

O(5A) N(5A) O(6A) N(4A) O(4A) O(3A)

N(5B) O(6B) N(4B) O(4B) O(5B) O(3B)

Fig. 3. Mutual arrangement of molecules in the DTeFÐTTFÐDTeF triad in a stack of structure I. In atomic numbering, the letter A indicates the basis molecules and B refers to the acceptor molecule related to the basis molecule by the translation z. bonds of the OáááS and NáááS types with the participation H and N atoms are equal to 162.4° and 170.7°, and the of the S(3) and S(2) atoms of the tetrathiafulvalene N(1)···H(3') distance is 2.49 Å. Moreover, the Cl atom molecules, the O(2) atom of the NO2 group, and the of the chlorobenzene molecule is involved in the sec- N(3) atom of the CN group of the acceptor molecules. ondary interaction with the fluorene molecule. This The SáááO and SáááN distances are equal to 3.04 and 3.08 interaction is rather strong: the CláááN distance is Å, respectively. The O(2)áááS(3)ÐC(4'), N(3)áááS(2)Ð 3.14 Å, the angle at the Cl atom is 159.5°, and the angle C(1'), S(3)áááO(2)ÐN(4), and S(2)áááN(3)ÐC(17) angles at the N(6) atom is 105.7°. Since the Cl atom in this are 172.8°, 168.3°, 130.7°, and 173.8°. These parame- molecule is disordered over two positions, several alter- ters are geometrically favorable for the n–σ* and n–π* native descriptions of the crystal structure become pos- interactions. sible. For example, one view holds that the Cl atom is statistically disordered over two positions in each sheet. Each tetrathiafulvalene molecule is involved in one According to other variants, the packing is character- more weak interaction: the CH group participates in the ized by different types of local ordering within either formation of the hydrogen bond with the CN group of ribbons, or sheets, or domains. The crystal packing has the acceptor molecule that belongs to the adjacent four- defied adequate description only on the basis of X-ray component macrocycle of the second type. The diffraction experiment. However, this is of no funda- N(6)···H(6') distance is equal to 2.45 Å, and the mental importance, because even the aforementioned N(6)áááH(6')ÐC(6') and H(6')áááN(6)ÐC(18) angles are hydrogen bonds will suffice to link the ribbons into 162.5° and 161.2°, which is typical of hydrogen bonds sheets. involving nitrile groups. This macrocycle contains no cavity large enough to house the solvent molecule. Sol- Apart from the weak interaction within the sheet, vent molecules of the i type are located outside the rib- the Cl atom is involved in the intersheet weak interac- bons. Each of these molecules participates in the sheet tion with the N atom of the CN group in the 7-position formation through the CÐHáááNC hydrogen bonds of (see Fig. 7). The Cl(1A)áááN(6C) distance is 3.29 Å, and two centrosymmetrically related acceptor molecules the C(1''A)ÐCl(1A)áááN(6C) and Cl(1A)áááN(6C)Ð belonging to the adjacent ribbons. These hydrogen C(18C) angles are equal to 93.9° and 94.4°, respec- bonds have a conventional geometry: the angles at the tively. Figure 7 also shows the intersheet short contacts

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N(1B) N(2B)

C(16B) C(14B) C(15B)

N(1A) N(2A) S(3A) C(5'A) C(16A) C(6'A) C(8B) C(14A) C(15A) C(9B)

C(1B) C(10B) C(13B) C(7B) C(2B) C(4'A) N(6B) N(3B) S(4A) C(18B) C(17B) C(9A) S(2A) C(11B) C(12B) C(1A) C(10A) C(1'A) C(18A) C(8A) N(6A) C(4B) C(13A) N(3A) C(3'A) C(7A) C(6B) C(17A) C(3B) C(2A) C(5B) C(11A) C(12A) C(2'A) C(6A) C(5A) S(1A) C(3A) C(4A) O(3 ) N(5B) O(2B) N(4B) O(1B) B

O(4B) O(3A) N(5A) N(4A) O(1A) O(4A) O(2A)

Fig. 4. Mutual arrangement of molecules in the DDDFÐTTFÐDDDF triad in a stack of structure II. In atomic numbering, the letter A indicates the basis molecules and B refers to the acceptor molecule related to the basis molecule by the translation x. of the j-type solvate molecule, whose role is not con- SCHEMATIC REPRESENTATION ﬁned only to ﬁlling the macrocycle cavity. The OF THE STRUCTURE TYPES Cl(2A)áááN(2A) and Cl(2B)áááN(2B) distances are OF CHARGE-TRANSFER COMPLEXES 3.16 Å, and the angles at the Cl and N atoms are equal As follows from the above analysis of the crystal to 154.1° and 129.3°, respectively. For these intersheet packing in structures I and II, the stack …ADADAD…, secondary bonds, there can also arise different-type which is composed of the alternating donor and accep- crystal packings due to the disordering of the Cl atoms. tor molecules, is a typical structural motif of the If the Cl atoms of both solvate molecules are statisti- charge-transfer complexes under consideration. The cally disordered, the sheets are cross-linked at random …ADADAD… stacks built up of the acceptor and donor into a three-dimensional framework. In the case when molecules of different sizes can form several geneti- cally related packings. Scheme IV represents four vari- the elements of local ordering of the Cl atoms are ants of the supramolecular architecture of alternating present in the structure, there can arise double sheets or stacks in the structure of charge-transfer complexes a rather ordered cross-linking of the sheets. However, (the large and small rectangles stand for the large-sized the elucidation of the actual situation calls for further acceptor and small-sized donor molecules, respec- X-ray structure investigation with crystals grown using tively; ovals symbolize the molecules of the third com- solvents such as p-dichlorobenzene or toluene. ponent; and dashed lines show the stack orientation).

ab c d Scheme IV.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 258 KUZ’MINA et al.

C(5'B) N(3C) N(3 ) C(5'A) S(3B) C(6'B) D S(4A) O(1C) C(4'B) S(4B) O(1D) C(4'A) O(7A) O(7B) O(5A) N(6A) S(2B) O(5B) N(6B) S(1 ) C(1'A) A S(1B) O(8A) C(1'B) N(5B) C(2'A) C(3'B) H(3'B) C(2'B) N(4 ) H(2'A) A H(2'B) N(4B) C(16A) O(3A) O(3B) N(2A)

C(5'F) N(3A) C(6'F) N(3B) S(3F) O(1A) S(4F) C(4'F) O(1B) O(7E) O(7F) C(2E) N(6E) O(5E) S(2F) C(1'E) N(6F) N(5E) C(1'F) O(8 ) C(3'E) O(8E) C(1F) C(3'F) F C(2'F) H(3'F) H(3'E) N(4E) H(2'F) C(16F) O(3E) N(2E) N(2F)

Fig. 5. A sheet of the molecules linked through weak interactions in structure I. Letters in atomic numbering indicate different symmetry-related molecules.

In the close packing a, one stack is shifted relative metallic) conductivity [27]. Thus, the description of the to the other stack so that the large-sized acceptor mole- packings in the structure of charge-transfer complexes cules partially ﬁll the interstack space in the vicinity of with the use of the graphs considered has revealed their the small molecule in the adjacent stack. It is this genetic relationship and directed the way to the con- supramolecular architecture that is observed in the trolled modiﬁcation of the supramolecular architecture charge-transfer complex I. As was shown above, the of these complexes. This can be achieved either by displacement of the small molecules with respect to the varying the size and shape of the acceptor and donor large molecules in parallel planes does not affect the π molecules or by substituting the particular functional interaction between the acceptor and donor molecules. groups for the other groups with different geometric However, this displacement can result in packing b, constraints on the formation of secondary bonds. which is characterized by stacking not only of the type …ADADAD… but also of the type …AAA…, i.e., the The crystal lattice with packing d involves the third stacking of large-sized acceptor molecules. It is component. This packing is characteristic of the struc- expected that, in the limit, at a certain size ratio of ture of complex II when solvent molecules of the j type the acceptor and donor molecules, packing b will trans- are disregarded. Actually, these molecules have no form into packing c with separate molecular stacks effect on the stacking motif of the crystal, because they …AAAA… and …DDDD… . Packing c is especially are arranged at large angles with respect to the sheets desirable because it corresponds to the highest (i.e., considered above.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 SUPRAMOLECULAR ARCHITECTURE OF TWO CHARGE-TRANSFER COMPLEXES 259

N(6A) C(17A) S(2A) C(6'B) C(18A) N(3A) H(6'B) C(1'A) Cl(1A) C(15A) C(3'A) Cl(1C)

N(1A) H(3'A) C(3"A) H(6'B) C(1'C) C(3'D) N(1F) C(3'C) C(1'D) N(1 ) Cl(1B) G H(3' ) C C(3'F) C(15F) C(1'F) C(4'E) Cl(1D) H(6'E) C(15G) C(17F) S(3E) C(6'F) C(4'F) S(2F) C(6'E) H(6'F) N(3F) N(6F) C(5'E) C(18G) C(18F) S(3F) N(6G) C(5'F) O(2B) Cl(2A) N(4F) N(4B) O(2C) O(2F) N(4G) N(4C) Cl(2B) O(2G) N(6B) H(6'C) S(3C) C(17C) C(18B) N(6C) H(6'D) S(2C) C(4'C) C(6'C) S(3D) C(18C) N(3C) Cl(1 ) C(4'D) S(4C) E C(6'D) C(1'C) C(15B) C(1' ) H(3'F) E N(1B) Cl(1G) H(3'H) C(3'F) N(1C) C(3'E) H(3'E) C(1'G) C(3'H) N(1H) C(1'F) C(1'H) S(1H) C(3'H) Cl(1F) C(15H) C(15I) C(1'H) C(4'G) N(1I) Cl(1H) C(6'G) C(17H) S(3G) H(6'G) C(6'H) S(2H) C(5'G) H(6' ) C(4'H) N(6H) H N(3H) C(18H) C(18I) S(3H) C(5' ) N(6I) H O(2D) Cl(2C) N(4D) O(2E) N(4E) Cl(2D)

Fig. 6. A sheet of the molecules linked through weak interactions in structure II. Letters in atomic numbering indicate the different symmetry-related molecules.

Note that the presence of the i-type solvate mole- the controlled modification of the supramolecular cules in the stacks of complex II can hardly affect the architecture of charge-transfer complexes through the intrastack π–π interaction, because these molecules proper choice of the third component. possess weak acceptor properties and enter into the stack only around its periphery. However, the inference In principle, other variants of the supramolecular can be made that, in the case when a molecule similar architecture can also be proposed for charge-transfer in size and shape to the chlorobenzene molecule but complexes with a stoichiometry differing from 1 : 1. with the donor or acceptor properties is introduced into For example, the structure of complexes with the sto- the lattice, the properties of the charge-transfer com- ichiometry A : D = 1 : 2 can be characterized by plex can be modified significantly with no change in the packing d, in which the second donor molecule plays general packing motif. This opens up one more way to the role of the third component. It should be noted that

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 260 KUZ’MINA et al.

N(6C) C(18C)

Cl(1A) N(6A) C(15D) N(1D) H(3'A) C(18A) C(3'A) C(1'A) Cl(1B) C(3'B) C(1'B) H(3'B) N(1A) C(15A) C(18D) N(6D)

C(18B)

N(6B)

N(2B)

Cl(2B) C(16B)

C(4'B) C(4'A)

C(16A) Cl(2A)

N(2A)

Fig. 7. Weak interactions of the C6H5Cl molecules involved in the i and j systems. there can also occur more complicated situations, 8. P. Erk, H. Meixner, T. Metzenthin, et al., Adv. Mater. 3, which will be discussed in separate works. 311 (1991). 9. K. Sinzger, S. Hünig, M. Jopp, et al., J. Am. Chem. Soc. ACKNOWLEDGMENTS 115, 7696 (1993). 10. S. Hünig, J. Mater. Chem. 10, 1469 (1995). This work was supported by the Russian Foundation for Basic Research (project no. 01-03-32474) and the 11. G. R. Desiraju, Chem. Commun., 1475 (1997). Royal Society. 12. G. R. Desiraju, Angew. Chem., Int. Ed. Engl. 34, 2311 (1995). REFERENCES 13. G. R. Desiraju, in Comprehensive Supramolecular Chemistry, Ed. by D. D. MacNicols, F. Toda, and 1. V. Khodorhovsky and J. Y. Becker, in Organic Conduc- R. Bishop (Pergamon, New York, 1996), Vol. 6, pp. 1Ð tors: Fundamentals and Applications, Ed. by J.-P. Farges 22. (Marcel Dekker, New York, 1995). 2. J. Ferraris, D. O. Cowan, V. V. Walatka, and J. H. Perl- 14. G. R. Desiraju, Curr. Opin. Solid State Mater. Sci., 451 stein, J. Am. Chem. Soc. 95, 948 (1973). (1997). 3. M. R. Bryce, Chem. Soc. Rev. 20, 355 (1991). 15. J. D. Dunitz, in The Crystal as a Supramolecular Entity, 4. A. Krief, Tetrahedron 42, 1209 (1986). Ed. by G. R. Desiraju (Wiley, Chichester, 1996), pp. 1Ð 30. 5. K. Kobayashi and Y. Mazaki, J. Synth. Org. Chem. (Jpn.) 46, 639 (1988). 16. G. R. Desiraju, Crystal Engineering: The Design of 6. A. Aumüller and S. Hünig, Angew. Chem., Int. Ed. Engl. Organic Solids (Elsevier, Amsterdam, 1989). 23, 447 (1984). 17. D. D. Mysyk, I. F. Perepichka, and N. I. Sokolov, 7. S. Hünig, Pure Appl. Chem. 62, 395 (1990). J. Chem. Soc., Perkin Trans. 2, 537 (1997).

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18. D. D. Mysyk, N. M. Sivchenkova, V. É. Kampar, and 22. S. Horiuchi, H. Yamochi, G. Saito, et al., J. Am. Chem. O. Ya. Neœland, Izv. Akad. Nauk Latv. SSR, Ser. Khim., Soc. 118, 8604 (1996). 621 (1987). 23. I. F. Perepichka, L. G. Kuz’mina, D. F. Perepichka, et al., 19. I. F. Perepichka, D. D. Mysyk, and N. I. Sokolov, in Cur- J. Org. Chem. 63, 6484 (1998). rent Trends in Polymer Photochemistry, Ed. by 24. L. G. Kuz’mina, Koord. Khim. 25 (9), 643 (1999). N. S. Allen, I. R. Bellobono, M. Edge, and E. Selli (Ellis 25. A. Ellern, J. Bernstein, J. Y. Becker, et al., Chem. Mater. Horwood, London, 1995), pp. 318Ð327. 6, 1378 (1994). 20. Yu. P. Getmanchuk and N. I. Sokolov, in Fundamentals 26. W. F. Cooper, J. W. Edmonds, F. Wudl, and P. Coppens, of Optical Memory and Media (Vishcha Shkola, Kiev, Cryst. Struct. Commun. 3, 23 (1974). 1983), issue 14, p. 11. 27. F. Vögtle, in Supramolecular Chemistry: An Introduc- tion (Wiley, Chichester, 1991), pp. 290Ð312. 21. P. Strohriegl and J. V. Grazulevicius, in Handbook of Organic Conductive Molecules and Polymers, Ed. by H. S. Nalwa (Wiley, Chichester, 1997), Vol. 1, p. 553. Translated by O. Borovik-Romanova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 262Ð267. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 297Ð302. Original Russian Text Copyright © 2002 by Lysenko, Kuznetsov, Medvedev, Zaœtsev. STRUCTURE OF ORGANIC COMPOUNDS

Molecular Structure of Diantipyrylphenylmethane and Its Complex with Neodymium Nitrate K. A. Lysenko*, M. L. Kuznetsov**, Yu. N. Medvedev**, and B. E. Zaœtsev*** * Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 117813 Russia ** Moscow State Pedagogical University, ul. Malaya Pirogovskaya 1, Moscow, 119882 Russia *** Russian University of Peoples’ Friendship, ul. Miklukho-Maklaya 6, Moscow, 117198 Russia Received December 13, 2000

Abstract—The structures of diantipyrylphenylmethane (DAPM) and its complex with neodymium nitrate ⋅ ⋅ ⋅ [Nd(NO3)3 DAPM CH3OH] 2CH3OH are determined by X-ray diffraction. The free ligand adopts a trans conformation with the opposite orientation of the oxygen atoms of the carbonyl groups. In the complex, diantipyrylphenylmethane acts as a bidentate ligand and coordinates the Nd atom through the carbonyl oxygen atoms, thus forming the eight-membered metallocycle. The coordination number of neodymium is nine (six O atoms of the bidentate nitrate groups, two O atoms of diantipyrylphenylmethane, and one O atom of the methanol molecule). © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION in the precipitation of the single crystals suitable for It is well known that pyrazolone-5 derivatives (spe- X-ray structure analysis. The crystal sample used for cifically diantipyrylmethane and its substituted deriva- data collection was sealed in a glass capillary. The crys- tives) and their complexes with d and f elements are tal data and main refinement parameters for compounds I widely used in pharmacology, analytical chemistry [1], and II are summarized in Table 1. Both structures were and dye chemistry [2]. Earlier [3], we reported the prep- solved by the direct method, and the non-hydrogen aration and spectral properties of the diantipyrylphe- atoms were refined first in the isotropic approximation nylmethane (DAPM) complexes with anhydrous lan- and then in the anisotropic approximation with the full- thanide nitrates. It was concluded from the IR spectro- matrix least-squares procedure. In structure I, the scopic data that the ligand bidentately coordinates a hydrogen atoms were located from the electron-density lanthanide atom through the oxygen atoms of the car- difference syntheses and refined isotropically. In struc- bonyl groups. However, analysis of the vibrational ture II, the positions of the hydrogen atoms were calcu- spectra cannot provide answers to questions as to which lated from geometric considerations and refined within conformation is adopted by the ligand in the complex the riding model. The hydrogen atoms of the coordi- and which structure (mononuclear or bridging) is nated and solvate methanol molecules in structure II formed. could not be included in the refinement, because they were not located. The absolute structure of II was deter- Moreover, the structural data on diantipyrylmethane mined by the refinement of the Flack parameter [8], derivatives and their complexes are few in number whose value was equal to Ð0.04(3). All the calculations [4−6]. In this respect, the aim of the present work was were performed with the SHELXTL PLUS 5 program to investigate the molecular structures of dian- package [9] on a personal computer. The atomic coor- tipyrylphenylmethane (I) and its complex with neody- dinates for structures I and II are listed in Tables 2 and mium nitrate (II) and to analyze the structural changes 3, respectively. in the ligand upon its coordination. RESULTS AND DISCUSSION EXPERIMENTAL General views of molecules I and II with atomic Single crystals I suitable for the X-ray diffraction numbering are shown in Figs. 1 and 2, respectively. study were prepared by recrystallization of DAPM from The bond lengths and angles in molecule I are close absolute methanol at room temperature. Complex II to the corresponding parameters in the other pyra- was synthesized by the reaction between methanol zolone-5 derivatives [10, 11]. In the crystal, the mole- × solutions of Nd(NO3)3 (0.3 M) and of the ligand (4 cule adopts a trans conformation (Fig. 1) with the 10–2 M) in a 1 : 1 stoichiometric ratio [3]. The evapora- opposite orientation of the carbonyl groups with respect tion of the solution at room temperature in a dry box to the plane of the C(2)ÐC(4)ÐC(6) fragment. This ori- (P2O5 served as a drying agent) for three days resulted entation is characterized by the O(1)C(1)C(5)O(2)

MOLECULAR STRUCTURE OF DIANTIPYRYLPHENYLMETHANE 263

Table 1. Crystal data and reﬁnement parameters for structures I and II Compound III

Formula C29H28N4O2 C32H31N7O14Nd Molecular weight 464.55 881.88

Space group P21/cP212121 a, Å 18.152(4) 10.899(3) b, Å 9.665(2) 13.031(3) c, Å 14.361(3) 27.377(6) β, deg 101.71(3) V, Å3 2467.0(9) 3888(2) Z 44 F(000) 984 1776 ρ Ð3 calcd, g cm 1.251 1.506 Diffractometer CAD-4 Enraf-Nonius Siemens P3 Radiation MoKα (λ = 0.71073 Å) MoKα (λ = 0.71073 Å) µ, cmÐ1 0.8 99.00 Absorption correction None DIFABS [7]

Tmin/Tmax 0.776/0.845 Temperature, K 293(2) 293(2) Scan mode θ/5/3θθ/2θ θ 2 max, deg 50 52 Number of unique reflections 3530 4095 σ R1 [on F for reflections with I > 2 (I)] 0.0364 (2193 reflections) 0.0425 (3177 reflections) wR2 (on F2 for all reflections) 0.1253 (3480 reflections) 0.1447 (4045 reflections) Number of parameters refined 429 487 pseudotorsion angle equal to 158.3°. The angle between molecule, these angles are 37.7° and 40.1° [11]. The the planes of two pyrazolone rings in molecule I N(2)ÐC(12) and N(4)ÐC(18) bond lengths [1.424(3) is 66.1°. The bond lengths in two antipyrine fragments and 1.420(3) Å, respectively] are close to the standard [C(1)C(2)C(3)N(1)N(2) and C(5)C(6)C(7)N(3)N(4)] value for the C –N 3 bond (1.42 Å [12]). This also of the DAPM molecule have close values (Table 4). ar sp indicates the lack of substantial conjugation between The nitrogen atoms in molecule I have the pyrami- the phenyl and pyrazolone rings. dal environment. The phenyl and methyl substituents at the nitrogen atoms are located on opposite sides of the Note that the bond lengths in two C=CÐC(O)ÐN(Ph) plane of the pyrazolone ring; the C(9)N(1)N(2)C(12) fragments of the N-phenyl substituted antipyrine rings and C(11)N(3)N(4)C(18) torsion angles are equal to differ slightly from one another. For example, the C(1)Ð Ð72.1° and Ð55.8°, respectively. Note that, for the N(1) O(1) bond [1.225(3) Å] is slightly shorter than the and N(3) nitrogen atoms, the mean deviation (0.37 Å) C(5)–O(2) bond [1.236(3) Å], and the N(4)–C(5) from the plane passing through the atoms bound to N(1) bond [1.388(3) Å] is slightly shorter than the N(2)–C(1) and N(3) is slightly larger than the mean deviation for [1.399(3) Å]. The C(2)ÐC(3) and C(6)ÐC(7) bond the N(2) and N(4) atoms (0.19 Å). The flattening of the lengths in the two rings are equal to each other atomic coordination observed for the N(2) and N(4) [1.350(3) Å]. Therefore, the splitting of the ν(CO) atoms is possibly due to their participation in the con- absorption band in the IR spectrum of I (1664 and jugation with the carbonyl groups. The NÐN bond 1644 cmÐ1) [3] can be attributed to the nonequivalence length in both heterocycles is virtually the same and of the C=O groups in the DAPM molecule. averages 1.411 Å, which is smaller than the standard Npyramidal–Npyramidal bond length (1.45 Å [12]). The dihe- The CÐCÐC angles at the C(4) atom are slightly dral angles between the planes of the phenyl and pyra- larger than the tetrahedral angle (112.1°Ð115.7°), and zole rings in molecule I are 35° and 50.3°. For compar- the C(2)C(4)C(24)C(25) and C(6)C(4)C(24)C(25) tor- ison, in the antipyrine molecule, the corresponding sion angles are equal to Ð146.7° and Ð12.9°, respec- angle is equal to Ð52.1° [10], and in the amidopyrine tively.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 264 LYSENKO et al.

× 4 2 × 3 Table 2. Coordinates ( 10 ) and equivalent isotropic thermal parameters (Ueq, Å 10 ) of the non-hydrogen atoms in structure I

Atom xyzUeq Atom xyzUeq O(1) 6070(1) 3495(2) 5745(1) 55(1) C(13) 4405(1) 3061(3) 5159(2) 47(1) O(2) 8463(1) Ð119(2) 7796(1) 45(1) C(14) 3765(2) 2968(3) 4459(2) 57(1) N(1) 5472(1) 900(2) 7086(2) 45(1) C(15) 3388(2) 1730(3) 4279(2) 63(1) N(2) 5333(1) 1986(2) 6415(2) 42(1) C(16) 3662(2) 568(3) 4785(2) 56(1) N(3) 8395(1) 880(2) 5456(1) 41(1) C(17) 4308(1) 636(3) 5476(2) 47(1) N(4) 8695(1) 93(2) 6271(2) 41(1) C(18) 9056(1) Ð1183(2) 6160(2) 42(1) C(1) 6014(1) 2608(2) 6337(2) 38(1) C(19) 9685(1) Ð1556(3) 6816(2) 52(1) C(2) 6582(1) 1955(2) 7064(2) 35(1) C(20) 10040(2) Ð2794(3) 6716(3) 65(1) C(3) 6237(1) 974(2) 7493(2) 40(1) C(21) 9777(2) Ð3638(3) 5964(3) 70(1) C(4) 7415(1) 2307(2) 7299(2) 32(1) C(22) 9155(2) Ð3271(3) 5297(3) 72(1) C(5) 8355(1) 469(2) 7016(2) 34(1) C(23) 8778(2) Ð2047(3) 5400(2) 59(1) C(6) 7875(1) 1624(2) 6660(2) 32(1) C(24) 7552(1) 3865(2) 7472(2) 32(1) C(7) 7939(1) 1862(2) 5754(2) 36(1) C(25) 8158(1) 4539(3) 7229(2) 48(1) C(8) 6547(2) Ð15(3) 8286(3) 56(1) C(26) 8295(2) 5930(3) 7452(2) 56(1) C(9) 4949(2) 902(5) 7754(3) 68(1) C(27) 7835(2) 6656(3) 7918(2) 50(1) C(10) 7605(2) 2963(3) 5077(3) 52(1) C(28) 7237(1) 5989(2) 8179(2) 49(1) C(11) 8925(2) 1236(4) 4854(3) 65(1) C(29) 7097(1) 4614(2) 7951(2) 41(1) C(12) 4679(1) 1893(2) 5678(2) 39(1)

The positions of the carbonyl groups allow the par- approaches the H(10a) atom [O(1)áááH(10a), 2.26 Å; ticipation of the oxygen atoms in the weak intramolec- C(10)···O(1), 3.167(3) Å; and the C(10)–H(10a)áááO(1) ular CÐHáááO contacts: the O(2) atom approaches the angle, 141(1)°]. H(4) atom attached to C(4) [O(2)···H(4), 2.49(2) Å; According to the X-ray structure analysis, the com- O(2)···C(4), 2.463(3) Å; and the C(4)–H(4)···O(2) position of complex II is represented by the formula ° ⋅ ⋅ ⋅ angle, 76(1) ], which results in the formation of the pla- [Nd(NO3)3 DAPM CH3OH] 2CH3OH. The coordi- nar H-bonded ﬁve-membered ring; and the O(1) atom nation sphere of the metal atom (the coordination num-

C(27) C(26) C(28) C(25) C(29) C(10) C(24) O(1) C(11) C(7) N(3) C(1) C(13) C(14) C(4) N(2) C(6) N(4) C(2) C(12) C(5) C(3) C(19) O(2) C(18) N(1) C(9) C(15) C(8) C(20) C(17) C(23) C(16)

C(21) C(22)

Fig. 1. A general view and the atomic numbering in molecule I.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 MOLECULAR STRUCTURE OF DIANTIPYRYLPHENYLMETHANE 265

× 4 2 × 3 Table 3. Coordinates ( 10 ) and equivalent isotropic thermal parameters (Ueq, Å 10 ) of the non-hydrogen atoms in structure II

Atom xyzUeq Atom xyzUeq Nd 6260(1) 9770(1) 1228(1) 34(1) C(8) 10807(9) 12153(9) 38(4) 55(3) O(1) 6917(5) 11297(4) 856(2) 38(1) C(9) 10086(11) 13772(8) 785(4) 70(3) O(2) 7816(6) 8981(5) 777(2) 46(2) C(10) 11045(8) 9511(7) Ð463(3) 51(2) O(3) 5509(7) 10790(6) 1921(2) 64(2) C(11) 11431(12) 7553(10) 113(5) 80(4) O(4) 5612(9) 7912(6) 1119(3) 83(3) C(12) 8060(8) 12984(7) 1401(3) 37(2) O(5) 4482(9) 8775(7) 1588(4) 102(4) C(13) 8881(11) 12995(8) 1787(3) 61(3) O(6) 4007(12) 7214(8) 1428(5) 146(5) C(14) 8480(13) 13392(10) 2232(4) 74(4) O(7) 7301(9) 8930(7) 1950(3) 83(3) C(15) 7302(16) 13791(11) 2275(5) 92(5) O(8) 8235(7) 10229(8) 1674(3) 75(2) C(16) 6537(12) 13741(11) 1900(5) 85(4) O(9) 9070(10) 9365(11) 2253(4) 142(5) C(17) 6897(11) 13354(9) 1449(4) 62(3) O(10) 4283(7) 10637(7) 938(3) 71(2) C(18) 9775(10) 7496(7) 968(3) 48(2) O(11) 5150(8) 9685(7) 401(3) 77(2) C(19) 10785(10) 7552(9) 1272(4) 66(3) O(12) 3473(8) 10500(10) 222(4) 113(4) C(20) 10891(17) 6860(11) 1652(5) 93(5) N(1) 9563(7) 12739(6) 738(3) 42(2) C(21) 9993(16) 6159(12) 1742(5) 92(5) N(2) 8425(6) 12548(6) 942(2) 38(2) C(22) 8994(15) 6118(10) 1434(5) 89(4) N(3) 10589(7) 8379(6) 245(3) 44(2) C(23) 8870(13) 6793(8) 1051(4) 67(3) N(4) 9674(7) 8210(6) 573(3) 44(2) C(24) 8401(7) 10768(7) Ð558(3) 35(2) N(5) 4666(10) 7952(8) 1380(4) 72(3) C(25) 8172(9) 9973(6) Ð877(3) 49(2) N(6) 8207(10) 9517(10) 1958(4) 78(3) C(26) 8033(11) 10129(9) Ð1369(3) 60(3) N(7) 4295(8) 10289(9) 521(4) 70(3) C(27) 8158(11) 11112(10) Ð1555(3) 65(3) C(1) 7910(7) 11700(6) 716(3) 32(2) C(28) 8363(9) 11934(7) Ð1237(4) 55(2) C(2) 8718(9) 11433(6) 324(3) 34(2) C(29) 8464(8) 11744(7) Ð739(3) 40(2) C(3) 9685(8) 12089(7) 349(3) 39(2) C(30) 6116(18) 11096(12) 2365(5) 112(6) C(4) 8432(6) 10528(6) 3(3) 29(2) C(1S) 1113(16) 10523(14) 1239(4) 113(5) C(5) 8788(9) 8947(6) 516(3) 34(2) O(1S) 1108(12) 10452(14) 1727(5) 182(7) C(6) 9162(7) 9587(6) 128(3) 31(2) C(2S) 3197(10) 11131(8) 2133(3) 100(3) C(7) 10265(8) 9218(7) Ð39(3) 42(2) O(2S) 2893(21) 11825(16) 2518(7) 141(8) ber is nine) is formed by three bidentate nitrate groups, Nd–O bonds are longer [2.528(8)–2.569(7) Å]. A NO3 two oxygen atoms of the carbonyl groups of DAPM, similar situation is observed for many of the nitrate whose coordination results in the closure of the eight- complexes [14, 15]. The structure parameters of the membered metallocycle, and the methanol molecule nitrate groups, which form virtually planar chelate (Fig. 2). Thus, our earlier conclusion concerning the rings with the neodymium atom, are similar to those bidentate coordination of DAPM in lanthanide com- described in [13Ð15]. The NdÐOMeOH bond length is plexes, which was made based on the analysis of the IR 2.357(7) Å. spectra [3], is conﬁrmed by the X-ray diffraction study. The coordination of DAPM results in considerable It should be noted that the lanthanide complexes structural changes in the ligand. First, the DAPM mol- with bidentate nitrate groups are characterized by the ecule undergoes a conformational transformation: the largest coordination numbers, which, apparently, can molecule changes over from the trans conformation in be explained by the forced shortening of the OáááO dis- the free ligand to the cis conformation with the tance in the nitrate group to 2.14 Å [13]. If the bidentate O(1)C(1)C(5)O(2) pseudotorsion angle equal to Ð13.8°. nitrate groups are treated as pseudomonodentate The angle between the planes of two pyrazolone rings ligands, the coordination polyhedron of the neody- in the coordinated DAPM molecule decreases mium atom can be described as a severely distorted only slightly (from 61.6° in the free ligand to 56.0° in octahedron. the complex). The bond lengths in the The NdÐO distances vary in the range 2.335(6)Ð C(1)C(2)C(3)N(1)N(2) and C(5)C(6)C(7)N(3)N(4) 2.569(7) Å: the Nd–ODAPM distances are the shortest antipyrine fragments of complex II differ insigniﬁ- among them [2.335(6) and 2.348(6) Å], whereas the cantly, as is the case in the uncoordinated ligand.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 266 LYSENKO et al.

C(27)

C(28) C(26)

C(29) C(25)

C(24) C(10) C(8) C(4) C(6) C(7) C(3) C(2)

N(1) O(12) N(3) C(9) N(7) C(11) O(10) O(11) C(5) N(2) C(1) O(1) N(4) C(17) O(2) C(12) O(4) Nd C(18) C(23) C(16) O(6) C(13) O(3) N(5) O(5) C(15) O(8) C(19) C(22) C(14) N(6) O(7) C(30) C(20) C(21) O(9)

Fig. 2. A general view and the atomic numbering in molecule II.

Table 4. Selected bond lengths d (Å) and angles ω (deg) in The coordination of the Nd atom in DAPM leads to structure I a lengthening of the CÐO carbonyl bonds [C(1)ÐO(1), 1.26(1) Å and C(5)–O(2), 1.28(1) Å] and a significant Atoms d Atoms d shortening of the N–N bonds [N(1)–N(2), 1.38(1) Å O(1)ÐC(1) 1.225(3) N(3)ÐC(11) 1.459(4) and N(3)–N(4), 1.36(1) Å]. Note that, despite the difference in the C(1)ÐO(1) and C(5)ÐO(2) bond lengths O(2)ÐC(5) 1.236(3) N(4)ÐC(5) 1.388(3) in the coordinated ligand, the IR spectrum of this com- N(1)ÐC(3) 1.396(3) N(4)ÐC(18) 1.420(3) plex exhibits a single ν(CO) band with a maximum at N(1)ÐN(2) 1.413(3) C(1)ÐC(2) 1.455(3) 1611 cmÐ1. It is possible that the splitting of the band is N(1)ÐC(9) 1.479(4) C(2)ÐC(3) 1.350(3) not observed because of the considerable width of this N(2)ÐC(1) 1.399(3) C(2)ÐC(4) 1.521(3) band. N(2)ÐC(12) 1.424(3) C(4)ÐC(6) 1.512(3) The coordination of the DAPM molecule results N(3)ÐC(7) 1.383(3) C(5)ÐC(6) 1.445(3) also in a significant flattening of the configuration of the bonds involving the nitrogen atoms. The deviations of N(3)ÐN(4) 1.409(3) C(6)ÐC(7) 1.350(3) the N(1) and N(2) atoms from the planes passing Atoms ω Atoms ω through the adjacent atoms are 0.12 and 0.22 Å, respectively. The corresponding deviations of the N(3) and C(3)ÐN(1)ÐN(2) 105.8(2) C(7)ÐN(3)ÐN(4) 105.6(2) N(4) atoms are 0.17 and 0.06 Å, respectively. Note that, C(3)ÐN(1)ÐC(9) 116.3(2) C(7)ÐN(3)ÐC(11) 122.0(2) similar to the uncoordinated ligand, the phenyl and N(2)ÐN(1)ÐC(9) 113.1(2) N(4)ÐN(3)ÐC(11) 115.4(2) methyl substituents in both substituted antipyrine C(1)ÐN(2)ÐN(1) 109.6(2) C(5)ÐN(4)ÐN(3) 109.9(2) fragments are located on opposite sides of the pyrazolone ring. The C(9)N(1)N(2)C(12) and C(1)ÐN(2)ÐC(12) 125.7(2) C(5)ÐN(4)ÐC(18) 126.9(2) C(11)N(3)N(4)C(18) torsion angles are Ð38.2° and N(1)ÐN(2)ÐC(12) 117.9(2) N(3)ÐN(4)ÐC(18) 119.0(2) 31.4°, respectively. A comparison of the bond length

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 MOLECULAR STRUCTURE OF DIANTIPYRYLPHENYLMETHANE 267 distributions in the pyrazolone heterocycles of the free 2. N. V. Trofimov, N. A. Kanaev, and A. I. Busev, Zh. and coordinated DAPM molecules suggests that the Obshch. Khim. 45 (2), 385 (1975). coordination of the DAPM molecule enhances the con- 3. M. L. Kuznetsov, Yu. N. Medvedev, B. E. Zaœtsev, et al., jugation of the nitrogen atoms with the C=CÐC=O frag- Zh. Neorg. Khim. 40 (2), 259 (1995). ment. This leads to a flattening of the environment of 4. V. K. Bel’skiœ, D. I. Semenishin, V. I. Yarovets, and the nitrogen atoms, a shortening of the NÐC bonds O. I. Kuntyœ, Kristallografiya 32 (4), 1029 (1987) [Sov. (1.37–1.39 Å), and an additional lengthening of the Phys. Crystallogr. 32, 605 (1987)]. C=C [1.36(1) and 1.37(1) Å] and C=O bonds. 5. Yu. N. Mikhaœlov, G. M. Lobanova, A. S. Kanishcheva, et al., Zh. Neorg. Khim. 28 (8), 2056 (1983). The lengths of the NÐC bonds between the pyra- 6. E. Akgun and U. Pindur, Arch. Pharm. (Weinheim) 317 zolone and phenyl rings and the CÐC bonds at the C(4) (9), 737 (1984). atom change insignificantly. This indicates that, in the 7. N. Walker and D. Stuart, Acta Crystallogr., Sect. A: DAPM molecule involved in the complex, there occurs Found. Crystallogr. A39, 158 (1983). virtually no conjugation between the pyrazolone rings 8. H. D. Flack, Acta Crystallogr., Sect. A: Found. Crystal- nor between the phenyl and pyrazolone rings. The logr. A39, 876 (1983). angles between the planes of the phenyl and pyrazolone ° ° 9. G. M. Sheldrick, SHELXTL, Version 5: Software Refer- rings in the complex (52.8 and 57.0 ) are smaller than ence Manual (Siemens Industrial Automation Inc., Mad- those in the free ligand. ison, 1994). 10. T. P. Singh and M. Vijayan, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. B29 (4), 714 (1973). ACKNOWLEDGMENTS 11. T. P. Singh and M. Vijayan, Acta Crystallogr., Sect. B: This study was supported by the Russian Founda- Struct. Crystallogr. Cryst. Chem. B32 (8), 2432 (1976). tion for Basic Research (project no. 00-03-32807) and 12. F. H. Allen, O. Kennard, D. G. Watson, et al., J. Chem. the State Program of Support for Leading Scientific Soc., Perkin Trans. 2, S1 (1987). Schools of the Russian Federation (project no. 00-15- 13. A. Wells, Structural Inorganic Chemistry (Clarendon, 97359). Oxford, 1984; Mir, Moscow, 1987), Vol. 2. 14. L. A. Aslanov, L. I. Soleva, M. A. Poraœ-Koshits, and S. S. Goukhberg, Zh. Strukt. Khim. 13 (4), 655 (1972). REFERENCES 15. W. T. Carnall, S. Sigel, and J. R. Ferraro, Inorg. Chem. 12, 560 (1973). 1. Diantipyrylphenylmethane and Its Homologs as Analyt- ical Reagents, Ed. by S. I. Gusev (Permsk. Univ., Perm, 1974). Translated by I. Polyakova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

CRYSTAL CHEMISTRY

General Structural Features of Centric and Acentric Polymorphic Pairs of Organic Molecular Crystals: I. Polymorphic Pairs P21/cÐP212121 L. N. Kuleshova and M. Yu. Antipin Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, GSP-1, 117813 Russia e-mail: [email protected] Received June 27, 2001

Abstract—A systematic investigation into the general structural features of centric and acentric polymorphic pairs of organic molecular crystals is performed using the data available in the Cambridge Structural Database. The main regularities of the formation of these modiﬁcations are revealed. The inference is made that the intermolecular association plays an important role in the formation of crystal packing. © 2002 MAIK “Nauka/Inter- periodica”.

INTRODUCTION the thermodynamic approach, provides an explanation of the existence of different polymorphic systems. In Investigation into the general structural features of the case when the above differences in free energies crystalline materials is one of the most important direc- increase, the possibility of forming polymorphic modi- tions of the engineering of organic molecular crystals fications decreases. At the same time, the real crystal [1]. This field of research involves the design of organic structure does not necessarily possess a minimum crystalline materials with a specified molecular energy, and, in some cases, another polymorphic mod- arrangement responsible for many physical solid-state ification can be obtained only through the change in characteristics, such as electromagnetic and nonlinear crystallization conditions. In 1965, McCrone [8] stated optical properties, electronic and ionic conductivities, that the number of polymorphic modifications of a par- etc. Manifestation of certain properties (for example, ticular compound is in direct proportion to the time and nonlinear optical properties) requires a noncentrosym- money spent on their search. Ignoring this circum- metric arrangement of molecules in the condensed stance can lead to an underestimate of many promising phase [2]. Among the nonlinear optical materials but centrosymmetric materials, merely because nobody known to date, organic crystals, as before, remain the has made an attempt to obtain another (possibly, non- most interesting objects of investigation. centrosymmetric) modification of the compound under As a rule, the search for new organic optical nonlin- investigation. ear materials involves a combination of two key stages. It should be noted that the existence of polymorphic The first stage includes the design and synthesis of sta- modifications is governed not only by the minimum ble chromophore molecules with high molecular hyper- free energy of the crystalline phases but also by the polarizability (molecular nonlinear optical engineer- kinetic processes of crystal nucleation and growth. ing). The second stage consists in preparing noncen- Hence, there is a need to use an approach that makes it trosymmetric crystals of chromophores with an possible to evaluate how the crystallization conditions optimum molecular orientation required to achieve the affect the formation of the crystal structure. One way of maximum macroscopic nonlinear optical effect (non- obtaining this estimate is to treat the crystallization pro- linear optical engineering of crystals). Currently avail- cess as a supramolecular synthesis in which the change able techniques of molecular nonlinear optical engi- in the physicochemical conditions brings about the for- neering have been worked out in sufficient detail [2–7]. mation of different packings. Within this approach, However, actual approaches to control over the molec- analysis of possible polymorphism is of crucial impor- ular packing in crystals of potentially important com- tance in designing new promising materials with a pounds are still being developed. The basic problems specified crystalline architecture. Among the most concerning the prediction and formation of optimum interesting works in the field of nonlinear optics, spe- crystal packing are associated with the great diversity cial mention should be made of research into the poly- of possible molecular orientations that are character- morphism of N-(2-acetamido-4-nitrophenyl)pyrroli- ized by minimum differences in free energies in the dine, 2-furaldehyde-5-nitro-N-benzoylhydrazone, and crystal. It is this circumstance that, in the framework of 2-(4-methoxyphenyl)-5-(4-nitrophenyl)pyrazolidin-5-ene

GENERAL STRUCTURAL FEATURES 269

[9]; 8-(4'-acetylphenyl)-1,4-dioxa-8-azaspiro[4,5]decane ent numbering in the Cambridge Structural Database. [10]; 5-nitro-2-thiophenecarboxaldehyde-4-methyl- According to McCrone [8], polymorphic modifications phenylhydrazone [11]; and 2-adamantylamino-5-nitro- are the crystals capable of forming the same compound pyridine [12]. However, similar works are still few in upon melting or dissolving. In the strict sense, poly- number. In this respect, it is of interest to reveal and morphic modifications are the crystals that belong only analyze the general structural features of polymorphic to the latter group of molecules. Since the potential modifications for all currently known compounds that nonlinear optically active molecules of interest fall in are capable of forming both centric and acentric crys- this group, we have restricted ourselves to the search talline modifications. In the future, this information can for polymorphic modifications with identical literal ref- be useful in designing crystalline materials with a spec- codes. Furthermore, we considered only the most com- ified molecular arrangement. monly encountered space groups [15], namely, the cen- First attempts to analyze the data accumulated in the trosymmetric space groups P1 , P21/c, and Pbca; the Cambridge Structural Database on compounds with chiral space groups P2 2 2 ,1 P2 , and P1; and the race- centric and acentric modifications were made by Brock 1 1 1 1 mic but noncentrosymmetric space groups Pc, Pna21, et al. [13] in 1991 and led the researchers to a number and Pca2 . For each of these space groups, we created of important conclusions. First, the universally 1 accepted concept that centrosymmetric (racemic) crys- a file containing the refcodes of all the compounds tals, as a rule, should possess higher densities and, specified by the attribute “form” in the qualifier. All the hence, should be more energetically favorable was dis- possible settings (including nonstandard settings) for proved. Second, the statistical analysis of the thermo- these groups were taken into consideration. Then, for dynamic characteristics for polymorphic modifications the purpose of revealing different structures with the existing at room temperature [14] revealed that the for- same literal refcode, the files of centrosymmetric space mation of centrosymmetric crystals are not thermody- groups were compared with the files of noncentrosym- namically preferable. On the one hand, the aforemen- metric space groups by using the specially developed tioned conclusions should lend impetus to researchers’ program REFCOMP. A total of 128 pairs of centric and efforts to obtain acentric crystalline modifications. On acentric modifications were found in the Cambridge the other hand, these inferences have failed to explain Structural Database (Version 2000). It should be noted that the predominance of centrosymmetric crystals among the pairs in which a crystal with the space group P21/c the structures collected in the Cambridge Structural serves as a centrosymmetric counterpart are consider- Database. ably more frequent (101 pairs) in the Cambridge Struc- tural Database as compared to those with the space In the present work, we undertook a systematic investigation into the structure of polymorphic (centric groups P1 (19 pairs) or Pbca (8 pairs). Possibly, this and acentric) modifications with the aim of revealing stems from the fact that the P21/c group is most often the general structural features that are essential to the encountered in organic compounds. For the refcodes understanding of the mechanisms of formation of the obtained in such a manner, all the data available in the crystal structure. Moreover, the available data on the Cambridge Structural Database were extracted through conditions and procedures of preparing polymorphs the database query QUEST3D. A number of polymor- were used to elucidate how these factors affect the for- phic modifications were rejected, because the available mation of crystals. In our opinion, the results of our data were either incomplete or invalid: errors in the investigation provide a better insight into the mecha- determination of the space group and atomic coordi- nisms of crystallization processes and can be useful in nates, the absence of atomic coordinates, etc. solving many problems of crystal engineering, including important problems concerning the design of acen- A preliminary analysis of the composition and tric crystalline modifications with nonlinear optical structure of the compounds found in the Cambridge properties. Structural Database demonstrated that, as would be expected, the majority of these compounds belong to planar conjugate or aromatic systems in which the THE CHOICE OF OBJECTS chirality arises as the result of insignificant rotations of Theoretically, centric and acentric crystalline modi- the functional groups with respect to each other. The fications of the same compound can be formed upon nonlinear optical activity is characteristic of eleven crystallization either from a racemic mixture of enanti- compounds, namely, MABZNA, MNPHOL, omers containing an asymmetric atom or from a solu- DEFDAN, BANGOM, FOVYOE, HAMNEO, tion containing achiral molecules or molecules with a CLBZNT, BAAANL, TALJIZ, NMBYAN, and low barrier to racemization. The former enantiomers SESHUT. The pairs of polymorphic modifications are referred to as resolved enantiomers and can have whose molecules belong to the group of resolved enan- different refcodes in the Cambridge Structural Data- tiomers are also encountered but considerably more base. The latter enantiomers are termed unresolved (or 1 This is the sole noncentrosymmetric space group in which the rapidly inverted) enantiomers and, as a rule, are antiparallel orientation of molecular dipoles excludes manifesta- described by identical literal refcodes but with a differ- tions of the nonlinear optical effect.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

270 KULESHOVA, ANTIPIN

Table 1. Total number of pairs of centric and acentric modiﬁcations available in the Cambridge Structural Database

P21/cÐP212121 41 P1 ÐP212121 9 PbcaÐP212121 2

P21/cÐP21 17 P1 ÐP21 4 PbcaÐP21 2

P21/cÐPna21 16 P1 ÐPna21 2 PbcaÐPna21 2

P21/cÐPca21 3 P1 ÐPca21 2 PbcaÐPca21 2

P21/cÐPc 11 P1 ÐPc 2 PbcaÐPc Ð

P21/cÐP14P1 ÐP1 3 PbcaÐP1 Ð rarely. For the most part, these modifications are found codes are assigned to the polymorphs formed simulta- to occur in the P21/c–P212121 pair. In the present work, neously. Otherwise, as was already mentioned, these we analyzed the general structural features of the polymorphs could be specified by different refcodes in P21/c–P212121 pairs of polymorphic modifications. The the Cambridge Structural Database and they would not other pairs listed in Table 1 will be considered in a sep- be included in the groups of compounds which we arate work. chose in this work. Polymorphic modifications of CHAPEP, BIGTIU, and SECZAB precipitated from different solvents (see Table 2). For the most part, the THE P21/c–P212121 POLYMORPHIC PAIRS molecules belonging to unresolved enantiomers are As was noted above, the most frequently encoun- rather bulky and do not form intermolecular hydrogen tered pair of polymorphic modifications embraces bonds. Therefore, the distinctive structural feature of 41 compounds with different chemical compositions. polymorphic modifications of this group is that the After rejecting the refcodes with incomplete data, we racemic pair of molecules and the pure enantiomer, as obtained 27 compounds (Table 2). Eighteen com- a rule, form topologically quite different structures in pounds that fall in the group of unresolved enantiomers which identical structural motifs are absent. The unit are presented in the first part of Table 2, and nine com- cell parameters of these polymorphic modifications dif- pounds that belong to the group of resolved enanti- fer noticeably; however, the unit cell volumes and den- omers, organometallic compounds, and complexes are sities of the polymorphs are found to be rather close in given in the second part of Table 2. Our main concern magnitude in the majority of the cases under consider- in the present work is with the former group of com- ation. pounds, because this group includes the majority of potential organic nonlinear optical materials. However, Polymorphic modifications of JAMLUE (Fig. 1) before proceeding further, we will dwell briefly on the whose molecules contain a hydroxyl group merit more most pronounced structural features of polymorphic detailed consideration. For this compound, a quite dif- modifications belonging to the latter group of com- ferent situation takes place. A system of van der Waals pounds. contacts and hydrogen bonds that arise between molecules of the same chirality through interactions of bridging iodide anions brings about the formation of Polymorphic Pairs distinct chiral layers. These layers form crystals of two of Resolved Enantiomers types, namely, the noncentrosymmetric crystals with The centric and acentric modifications, which the P212121 space group (congruent stacking of molec- belong to the group of resolved enantiomers (nine com- ular layers of the same chirality) and the centrosymmet- pounds), were synthesized under the same conditions ric crystals with the P21/c space group (stacking of (one pot polymorphism) in more than half of these molecular layers of different chiralities). In this case, cases (JAMLUE, FESKAP, CUMTAF, BARWUM, the unit cell parameters prove to be close in magnitude2 BUNKOK, and VISSOF). Hereafter, these polymor- and the packing densities are also very close in magni- phic modifications will be referred to as concomitant tude and differ by only 0.01 g/cm3. modifications. Under these conditions, the formation of racemic crystals is accompanied by the spontaneous 2 The unit cell parameters are considered to be close in magnitude separation of the enantiomers. Such a high percentage in the case when they differ by no more than 5Ð10%. The same is also true for the angle of monoclinicity, which, as a rule, is close of spontaneously separated enantiomers among the to 90°. The unit cell parameters are multiples when the asymmet- compounds under investigation can be a consequence ric part of the unit cell in one of the modifications under consider- of the procedure of their choice. Indeed, identical ref- ation involves several molecules (Z' > 1).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 GENERAL STRUCTURAL FEATURES 271 C ° C ° 1 2 1 = 140 = 25 2 ex ex 1 T T 2 C C ° ° P

C; C; ° °

c / ferent systems of hydrogen ferent systems of hydrogen = 295 K = 22 K, nonlinear optical K, = 22 = 102 = 114 = 73Ð74 = 84Ð85 1 2 m m ex m m ex Different positions of methoxy Different groups Identical systems of hydrogen Identical systems of hydrogen bonds Phase transition P bonds T T Phase transition (heating) (potential) T T material (potential) T T Solvent Comments -propanol n

Both from ethanol Dif Both from the mother solution, spontaneously Both from water Both from benzene and cyclohexane of Both from the reaction mixture Both from benzene Both from the ethyl acetateÐpetroleum ether mixture

geometry Molecular

Dif-

Close ferent ferent ferent fragments Packing

Dif-

Close Dif- ferent ferent parameters Unit cell Unit Dif- tiple Close Close Close Close Close Close Close Dif- Close Close Close Both through evaporation Close Close Close Close Close Dif- ferent 3 , g/cm D 3 , Å V ' 1 1340.81 1.640 1291.61 1.702 1596.51 1.191 1587.5 1.198 2 2329.32 1.205 2324.51 1.207 1854.51 1.162 1933.61 1.114 1347.54 1.327 Mul- 5362.91 1.3341 625.3 1.478 599.11 1.542 2194.81 1.266 2144.81 1.295 1231.12 1.279 2607.6 1.208 Nonlinear optical material Z 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 n c c c c c c c / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 group Space P P P P P P P P P P P P P P P P ) D ∆ ( (Ð0.029) (Ð0.016) (Ð0.07) (Ð0.002) (Ð0.007) (Ð0.064) (+0.048) (+0.071) CLBZAM DMFUSC DPGUAN DTBPTP MABZNA FIKFIO MNPHOL REFCODE 1 2 1 2 Concomitant polymorphic modiﬁcations of unresolved enantiomers and chiral molecules Concomitant polymorphic modiﬁcations of unresolved 3 3 1 2 P CH CH – COOH 3 3 c / 1 2 CH CH P Ph Ph 3 3 2 Compound NCH N OCONHC O OH CH CH 2 S CONH CH Polymorphic pairs CC Cl 2 2 tBu tBu Ph Ph NO NO Nimodipine* VAWWEV 2. Table

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 272 KULESHOVA, ANTIPIN 1 2 1 C C 2 ° ° 1 2 P C, nonlinear C ° °

c / ferent systems of = 313 = 310 = 237Ð238 = 239Ð240 1 2 m m m m T (potential) optical material Phase transition P bonds hydrogen T complex Charge-transfer Identical systems of bonds hydrogen T hydrogen bonds hydrogen bonds T Ð Solvent Comments

Hot ethanol Diluted ethanol Dif Petroleum ether Acetone Dichloroethane Aqueous solutions, kinetics different temperature SublimationBenzene, methanol Identical systems of Identical systems of etherÐbenzene

geometry Molecular

rent rent rent rent

Close Close Close Ð Diffe- Diffe- Diffe- Diffe-

fragments Packing

rent rent rent rent rent rent rent

Close Close Close Close Diffe- Diffe- Diffe- Diffe- Diffe- Diffe-

parameters Unit cell Unit rent rent rent rent rent rent tiple tiple Close Close Close Close Diffe- 3 , g/cm D 3 , Å V ' 3 2450.2 1.457 Mul- 1 625.3 1.478 Diffe- 1 990.2 1.3761 Diffe- 876.1 1.289 Mul- 1 1340.82 1.640 Diffe- 2353.5 1.3453 Diffe- 3491.21 1.497 Diffe- 1371.9 2.315 Diffe- 21 1624.0 1.465 823.6 1.445 1 599.1 1.5421 995.3 1.3692 1743.11 Ethanol at room 1.297 1180.7 1.4301 1279.7 Concentrated ethanol 1 1.206 1291.61 1.702 Acetone, ethanol 1154.8 acetate Ethyl 1.370 1 1177.9 Ethanol 1 1.479 1343.4 Acetic acid 2.364 Petroleum Z 1/2 1192.61/2 1.415 1278.2 1.207 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 c c n c c c n n c c / 2 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 group Space 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 P P P P P P P P P P P P P P P P P P P P P ) D ∆ ( (Ð0.049) (Ð0.064) (Ð0.062) (Ð0.025) (Ð0.008) (Ð0.008) (Ð0.015) (+0.007) (+0.001) (+0.018) (+0.012) Polymorphic modifications of unresolved enantiomers prepared from different solvents enantiomers prepared from different Polymorphic modifications of unresolved REFCODE DEFDUN COYMOS BOPSAA HETPAL DPIPDS CIVTUM NOSWAR PUMVIC CLPHTE FABFUJ OH 2 Polymorphic modifications of unresolved enantiomers prepared from the same solvent under different conditions under different enantiomers prepared from the same solvent Polymorphic modifications of unresolved CH 2 -di- 2 2 2 OH H 2 NO Compound NO ]pyran-6-one CONH CONH d CO O CH (Contd.) , N N CH CHOH b 2 NSSN N NH N Ph N Cl Cl Te Te Cl 2,3,8-Trinitro-6 benzo[ Table 2.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 GENERAL STRUCTURAL FEATURES 273 C C ° ° C ° C ° = 174Ð175 = 205Ð206 = 103 = 84 m m m m Identical systems of bonds hydrogen T T T T Solvent Comments 2 -Xylene

Both from the reaction mixture ether ethyl Methyl Both from the reaction mixture Both from the reaction mixture upon cooling From cooled methanol, spontaneously From the mother solution, spontaneously CS m From ethanol, spontaneously

geometry Molecular

rent rent rent rent rent rent rent rent

Diffe- Diffe- Diffe- Diffe- Diffe- Diffe- Diffe- Diffe-

fragments )-one.

H Packing

rent rent rent rent rent rent rent rent

Diffe- Diffe- Diffe- Diffe- Diffe- Diffe- Diffe- Diffe-

parameters Unit cell Unit rent rent rent rent rent rent rent rent Close Close Close 3 , g/cm D ][1,4]dibenzodiazepin-6(5 d 3 , Å V ' 11 1285.0 1293.6 1.587 1.577 1 1307.11 1.357 1304.6 Diffe- 1 1.360 1888.41 1.9691 1850.0 Diffe- 2603.5 2.010 1 1.3171 2032.6 Diffe- 1531.0 1.298 1 Water 2 1.636 1590.1 Diffe- 3274.11 1.575 1 1.471 1696.5 Diffe- 1808.91 1.418 2 1.422 1806.0 Diffe- 3622.91 1.424 1 1.277 1826.0 Diffe- 1372.11 1.267 1.202 1319.7 Diffe- 1.250 Z 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 -tetrahydrooxazolo[3,2- 1 1 1 1 1 1 1 1 1 c c c c c c c c c b / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 / 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 P P P P P P P P P P P P P P P P P P Space group ) D ∆ Polymorphic pairs of resolved enantiomers, organometallic compounds, and complexes enantiomers, organometallic Polymorphic pairs of resolved ( (Ð0.003) (Ð0.041) (Ð0.048) (Ð0.004) (+0.018) (+0.053) (+0.010) (+0.010) REFCODE -2,4,6,3,5-thiotriazadiphosphorine-1-oxide. VISSOF CHAPEP FESKAP CUMTAF SECZAB JAMLUE 6 λ (2-ﬂuorophenyl)-2,3,7,11 b 2 – I 3 3 3 NH CH CH CH N + + 2 OH 2 CH Compound CH C O II Ph (Contd.) H C OH As * Nimodipine is isopropyl-2-methoxyethyl-2,6-dimethyl-4-(3-nitrophenyl)-1,4-dihydro-3,5-pyridinedicarboxylate. ** Haloxazolam is 10-bromo-11 POC Ph Ph *** 1,3,3,5,5-penta(1-aziridinyl)-1- O O Haloxazolam****** BUNKOK BIGTIU Diethyl-dibenzoberrelene-11,12- dicarboxylate ***4,4-Diphenylcyclohexa-2-en-1-one BARWUM Table 2. **** 2,2,4,4,6,6-hexakis(1-aziridinyl)-cyclotri(phosphazene).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 274 KULESHOVA, ANTIPIN

(a) (a)

0 b 0 c

(b) (b)

0 c

b b 0

Fig. 1. Packings of identical chiral layers in the (a) P212121 Fig. 2. Projections of identical chiral layers in the and (b) P21/c modiﬁcations of JAMLUE. The layers are (a) P212121 and (b) P21/c modiﬁcations of DMFUSC onto arranged in the (110) plane. the xy plane.

Polymorphic Modifications centration, temperature, and crystallization rate. Poly- of Unresolved Enantiomers morphic modifications of BOPSAA, HETPAL, DPIPDS, CIVTUM, and CLPHTE were synthesized The centric and acentric modifications, which from different solvents. belong to the group of unresolved enantiomers, are also The majority of the molecules involved in this group frequently formed simultaneously (CLBZAM, are planar, conjugate, and small in size. The chirality of DMFUSC, DPGUAN, DTBPTP, MABZNA, these molecules stems from insignificant rotations of MNPHOL, FIKFIO, and VAWWEV). (The thermody- the functional groups with respect to each other without namic aspect of the concomitant polymorphism was appreciable expenditure of energy. It is worth noting discussed in detail in the review by Bernstein et al. that different polymorphic modifications of these com- [16].) In three other cases (FABFUJ, DEFDUN, and pounds exhibit an even more pronounced tendency for COYMOS), different modifications were prepared the unit cell parameters (and, correspondingly, the den- through crystallization from the same solvent but under sities of crystals) to retain the close values, provided different kinetic conditions by varying the initial con- that packing fragments remain unchanged.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 GENERAL STRUCTURAL FEATURES 275

(a)

b 0

0 b

a a

Fig. 3. Modiﬁcations of FIKFIO: (a) molecular geometries and molecular packings in the (b) P212121 and (c) P21/c modiﬁcations.

Concomitant polymorphic modifications. Identi- metric crystal, whereas the chains of different chirali- cal chiral packing fragments are most frequently ties are linked together to form a centrosymmetric observed in concomitant polymorphic modifications. structure with unit cell parameters that are close in For example, the CLBZAM polymorphs are character- magnitude. Identical stable chiral fragments are ized by identical chiral layers formed at the expense of revealed in other concomitant modifications of this the intermolecular hydrogen bonds NHáááO. Although group. The sole exception is represented by the FIKFIO the DMFUSC polymorphic modifications contain no polymorphic modifications (Fig. 3). These modifica- hydrogen bonds and the crystals are formed through tions involve different molecular associates and, hence, pure van der Waals interactions, these structures are are characterized by different packing densities and dif- also built up of identical chiral layers (Fig. 2). The ferent lattice parameters. In this case, the formation of structures of DMFUSC polymorphic modifications dif- different supramolecular fragments can be explained fer in their layer stacking: congruent stacking with a by the fact that molecules with different conformations shift is observed in the acentric modification and anti- already coexist in a solution of this compound. parallel stacking with inversion occurs in the centric A very interesting example of different packings modification. The polymorphic structural phase transi- α β formed by identical molecular associates is provided by tion (P21/c, Z' = 1) 383 ä (P212121, Z' = 1) the DPGUAN modifications. Both crystalline forms are between the α and β modifications of DMFUSC is characterized by Z' = 2; i.e., they are polysystem crys- observed at a temperature of 383 K. The phase transfor- tals. In both modifications, the chains with a similar mation is accompanied by the mutual cooperative rota- structure (Fig. 4) are composed of crystallographically tion of the layers with respect to each other by 30° and different molecules. Note that both chains are pseudo- the inversion of each second layer. A similar situation symmetric. In the noncentrosymmetric crystal, mole- arises with polymorphic modifications of VAWWEV. cules in the chain are related by an approximate glide- In these compounds, chains containing molecules of reflection plane along the c-axis. In the centrosymmet- the same chirality are formed through hydrogen bonds. ric crystal, molecules in the chain are related by the The chains of the same chirality form a noncentrosym- glide-reflection pseudoplane a, which approximately

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 276 KULESHOVA, ANTIPIN

(a) (b)

N(4A) N(4A)

Fig. 4. Structures of chains formed by independent molecules in the (a) P212121 and (b) P21/c modifications of DPGUAN in the projections parallel and perpendicular to the chain. coincides with the (1 1 0) plane. A detailed analysis of Polymorphic modifications prepared from the the molecular packing in these crystals according to the same solvent under different conditions. Identical scheme proposed in our earlier work [17] revealed that and stable supramolecular fragments can be formed not both pseudosymmetric crystals can be treated as the only in concomitant modifications but also in polymor- same hypothetical crystal with the pseudogroup of phs precipitated from the same solvent under different higher symmetry: [(P212121, Z' = 2) + c(1, 0.25, 1)] conditions by varying the concentration, temperature, or crystallization rate. An example of the kinetic poly- (Pbca, Z' = 1) [(P21/c, Z' = 2) + a(110)]. However, no real crystal with this symmetry was found. morphism is represented by the FABFUJ polymorphic modifications. The main crystallographic parameters of these modifications and the conditions of their preparation are given in Table 3. Modifications II and III belong to polysystem crystals and exhibit a pseudosymmetry (the pseudosymmetry was thoroughly considered in our recent work [17]). All three polymorphic modifications presented in Table 3 are built up of identical chiral layers (Fig. 5), which, in turn, are formed through hydrogen bonds. Note that these modifications 0 b differ only in the type of layer packing. A rapid cooling of the saturated solution leads to the formation of crystals with antiparallel packing and inversion, whereas the structure of crystals grown through evaporation from diluted solutions is characterized by congruent packing (Fig. 6). It is worth noting that modification II grown under virtually equilibrium conditions has the highest density and is completely ordered, whereas modifications I and III are partially disordered along the c-axis. a In some cases, especially when molecules are not involved in strong intermolecular hydrogen bonds, a Fig. 5. Structure of a chiral layer in the FABFUJ modifica- change in the crystallization temperature can bring tion. about a change in the orientation of substituents and,

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 GENERAL STRUCTURAL FEATURES 277

(a) (b) (c)

a a a

0 c 0 c 0 c

Fig. 6. Packings of chiral layers in the (a) P212121 (Z' = 1), (b) P212121 (Z' = 2), and (c) P21/c (Z' = 3) modiﬁcations of FABFUJ.

(a) (b)

0 c

Fig. 7. Molecular packings in the (a) P212121 and (b) P21/c modiﬁcations of DEFDUN in the projections parallel and perpendicular to the chain.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 278 KULESHOVA, ANTIPIN

(a)

(b)

Fig. 8. Structures of chains in the (a) P212121 and (b) P21/c modifications of BOPSAA. hence, in the type of supramolecular associate. For tric form is unstable at room temperature and, with example, thermodynamic polymorphism occurs in the time, undergoes the phase transition P21/c (Z' = 1) DEFDUN modifications. The orientation of the nitro P212121 (Z' = 1). This transition is attended by a change groups with respect to the central benzene ring in mol- in the orientation of the nitro groups with inversion and ecules of the centric modification, which was prepared insignificant rotations (by ~35°) of the molecules about in a hot supersaturated ethanol solution, differs from their centers of gravity. In turn, this leads to a transfor- that of the acentric modification synthesized at room mation of the system of short contacts. temperature. As a consequence, different systems of Polymorphic modifications synthesized in differ- short contacts stabilizing the structure (Fig. 7) are ent solvents. Identical supramolecular fragments (and, formed in these crystals. However, it can be seen from correspondingly, close values of unit cell parameters) Fig. 7 that the mutual arrangements of molecules in the are also observed in the polymorphic modifications layer differ only slightly and the layers themselves are produced from different solvents under different condi- composed of molecules of the same chirality. The cen- tions. This situation occurs, for the most part, when the

Table 3. Crystallographic parameters for polymorphic modiﬁcations of FABFUJ Modiﬁcation I II III

Space group P212121 P212121 P21/c Z'123 a 13.556 13.572 13.543 b 8.271 8.248 8.290 c 7.340 14.508 22.591 β 90 90 104.97 3 Dcalcd, g/cm 1.445 1.466 1.457 Crystallization conditions Evaporation of unsaturated Slow evaporation of diluted Rapid cooling of saturated aqueous solution aqueous solution aqueous solution

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 GENERAL STRUCTURAL FEATURES 279

(a) (b)

0 c 0 a

b b

Fig. 9. Molecular packings in the (a) P212121 and (b) P21/c modifications of HETPAL. supramolecular fragment is formed through sufficiently an appropriate solvent for crystallization of these mod- strong intermolecular hydrogen bonds, which are ifications were unsuccessful or that the second modifi- retained irrespective of the solvent type. The polymor- cation was simply overlooked. For example, the first phic modifications of BOPSAA (Fig. 8) and HETPAL data on the polymorphism in MNPHOL were reported (Fig. 9) can serve as examples. However, we cannot 120 years ago. However, the acentric form (in addition rule out the possibility that earlier attempts to choose to the well-known centric modification) was found among the MNPHOL crystals precipitated from benzene only in 1996. Molecules of all the other pairs of polymorphic modifications listed in Table 2 are incapa- ∆D, g/cm3 ble of forming strong intermolecular hydrogen bonds. In the case of different solvents, these molecules adopt different geometries; the molecular topology and unit cell parameters in polymorphic modifications are also different. The difference between the densities of cen- 0.05 tric and acentric modifications has a standard value, i.e., does not exceed 3Ð4%. Although the main objective of the present work was to analyze the general structural features of polymorphic modifications, brief mention should be made of the relative density of centric and acentric modifications. Figure 10 displays the differences ∆D between 0 the densities of centric and acentric modifications. The values of ∆D are equal, on average, to ~1Ð2% for the majority of the modifications under investigation and do not exceed 3Ð4% even for the largest differences that correspond to the polymorphic modifications of DTBPTP and MNPHOL, for which the densities were measured at different temperatures (see Table 2). The only exception is provided by the FIKFIO modifica- –0.05 tions whose densities differ by almost 6%. The close densities of different polymorphic modifications imply that their free energies are also rather close in magnitude. By and large, this confirms the inferences made earlier by Brock et al. [13]: compared to the acentric Fig. 10. Distribution of the ∆D values (∆D is the difference crystals, the centric crystals have a lower density and, between the densities of centric and acentric modification). hence, are less energetically favorable. Moreover, it

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 280 KULESHOVA, ANTIPIN should be noted that, in the pair of polymorphic modi- This work was supported by the Russian Foundation fications under consideration, the acentric crystals with for Basic Research, project nos. 00-03-32840a and a higher density are encountered even more frequently. 99-07-90133. This is especially true in regard to concomitant polymorphic modifications with close lattice parameters. REFERENCES 1. L. N. Kuleshova and M. Yu. Antipin, Usp. Khim. 68, 3 CONCLUSIONS (1999). In summary, we can draw the conclusion that, in all 2. Nonlinear Optical Properties of Organic Molecules and cases when the polymorphs are built up of identical sta- Crystals, Ed. by D. S. Chemla and J. Zyss (Academic, ble supramolecular fragments, the unit cell parameters New York, 1987), Vols. 1, 2. are either close in magnitude or multiples irrespective 3. S. R. Marder and J. W. Perry, Science 263, 1706 (1994). of the technique used for preparing the crystals. The 4. C. B. Gorman and S. R. Marder, Proc. Natl. Acad. Sci. examples considered above confirm the inferences USA 90, 11 297 (1993). made in [17] that the intermolecular association plays 5. G. Bourhill, L.-T. Cheng, G. Lee, et al., Mater. Res. Soc. an important role in the formation of crystal packing. In Symp. Proc. 328, 625 (1994). the case when molecules form stable associates through 6. S. R. Marder, C. B. Gorman, F. Meyers, et al., Science hydrogen bonds, the crystal structure is composed not 265, 632 (1994). of individual molecules but of larger-sized supramolec- 7. C. B. Gorman and S. R. Marder, Chem. Mater. 7, 215 ular fragments. The displacement of these fragments (1995). with respect to each other can lead both to a lowering of 8. W. C. McCrone, in Physics and Chemistry of the the symmetry of the real crystal and the appearance of Organic Solid State, Ed. by D. Fox, M. M. Labes, and a pseudosymmetry and to the formation of different A. Weissberger (Interscience, New York, 1965), Vol. 2, crystalline modifications. When molecules form stable p. 725. chiral associates, there arise centric and acentric poly- 9. S. R. Hal, P. V. Kolinsky, R. Jones, et al., J. Cryst. morphic modifications. This inference is in complete Growth 79, 745 (1986). agreement with the results obtained by Bernstein et al. 10. H. Kagawa, M. Sagawa, T. Hamada, et al., Chem. Mater. [18], who performed a numerical simulation of the pos- 8, 2622 (1996). sible crystal packings of achiral planar molecules of 11. F. Pan, C. Bosshard, M. S. Wong, et al., Chem. Mater. 9, aromatic hydrocarbons. In [18], it was demonstrated 1328 (1997). that a simple displacement of identical planar achiral 12. M. Yu. Antipin, T. V. Timofeeva, R. D. Clark, et al., layers by half the translation can readily result in the J. Mater. Chem. 11, 351 (2001). transformation from the centrosymmetric group P21/c 13. C. P. Brock, W. B. Schwezer, and J. D. Dunitz, J. Am. Chem. Soc. 113, 9811 (1991). to the chiral group P212121, and vice versa. In our opinion, these arguments are also true for molecules with a 14. A. Gavezzotti and G. Filippini, J. Am. Chem. Soc. 117, low racemization barrier. It is not improbable that the 12 299 (1995). above simplicity of the transformation also plays a role 15. C. P. Brock and J. D. Dunitz, Chem. Mater. 6, 1118 in the occurrence of this pair of centric and acentric (1994). modifications. 16. J. Bernstein, R. D. Davey, and J.-O. Henc, Angew. Chem. Int. Ed. Engl. 38, 3440 (1999). 17. L. N. Kuleshova, V. N. Khrustalev, and M. Yu. Antipin, ACKNOWLEDGMENTS Kristallografiya 45, 1034 (2000) [Crystallogr. Rep. 45, We are grateful to A.I. Yanovskiœ for the develop- 953 (2000)]. ment of the REFCOMP program and his assistance 18. J. Bernstein, J. A. Sarma, and A. Gavezzotti, Chem. with the Cambridge Structural Database and to Phys. Lett. 174, 361 (1990). V.N. Khrustalev and G.G. Andreev for their help in preparing the manuscript. Translated by O. Borovik-Romanova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

CRYSTAL CHEMISTRY

On Polymorphism and Morphotropism of Rare Earth Sesquioxides P. P. Fedorov, M. V. Nazarkin, and R. M. Zakalyukin Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia Received March 28, 2001

Abstract—The data on the phase transitions in the series of rare earth sesquioxides are considered in terms of the thermodynamic theory of morphotropism. It is shown that the scheme of polymorphism and morphotropism suggested by V.V. Glushkova is the most adequate. The derivatographic dehydration of lanthanum and neodymium oxide hydrates is performed. The transition of the cubic C modiﬁcation into the hexagonal A modiﬁcation of neodymium oxide is accompanied by the removal of volatile impurities and, thus, is not a polymorphic transition proper. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION thermodynamic function of a compound is a smooth Polymorphism, morphotropism, and crystal struc- function of ionic radius r [which is the necessary and tures of rare earth (RE) sesquioxides of the composition sufficient condition for the description of the corre- R O in oxidation degree III have been studied in [1Ð13]. sponding dependences of the phase-transition tempera- 2 3 tures T(r) (e.g., melting) by smooth functions], then the Three main structure types in which these compounds curve separating the regions of the thermodynamic sta- are crystallized at room temperature have been known bility of the phases having different structures in the since the studies of Goldschmidt: hexagonal A-type, the T–r coordinates should have a vertical tangent with the monoclinic B-type close to A, and the cubic C-type. At ° approach to the temperature of absolute zero. If the temperatures exceeding 2000 C, there exist two more Debye model is applicable to both modifications, then, modifications denoted as H and X. in the low temperature range, this curve is described by At present, two main schemes of polymorphism and the limiting equation morphotropism in the R2O3 series (Figs. 1a, 1b) are | |n known. In the scheme suggested by Foex and Traverse T = a r0 – r , (Fig. 1a), the existence range of the B modifications in the temperatureÐcationic radius diagram has a lenslike where a and r0 are the parameters and n = 0.25. For tri- shape and is limited from below by the existence range fluorides RF3, the experimental data yield n = 0.284 of the cubic C-phases. Moreover, it is also assumed that [15]. The aim of the present study is to consider the there exist low-temperature C-modifications of cerium polymorphism of RE oxides in terms of the thermody- oxides crystallizing in the A-type (R = LaÐNd). The namic theory of morphotropism. low-temperature phase transitions with the participation of C-modifications are indicated in accordance with data [1]. In the Glushkova scheme (Fig. 1b), the THERMAL DECOMPOSITION OF LANTHANUM boundaries between the fields of the A- and B- and AND NEODYMIUM HYDROXIDES between the B- and C-modifications are almost vertical in the low-temperature region. The reversible B C There are numerous data on the stabilization of the transitions take place only for the Tb, Dy, and Ho cubic C-modifications of RE oxides by various impuri- oxides. The low-temperature C-modifications of the ties and, first of all, by water [2, 4, 5]. We studied here LaÐGd oxides are interpreted as nonequilibrium ones. the dehydration of lanthanum and neodymium oxide hydrates by the derivatographic method (a Q-1500 deri- The canonical scheme suggested in [9, 11, 12] is vatograph, platinum crucibles, heating rate 10°C/min). widely used; however, it is somewhat erroneous. The starting materials were chemically pure La2O3 and The use of the thermodynamic theory of morphotro- Nd2O3 reagents kept in air for not less than twenty pism [14, 15] allows one to solve the problem, which years. has been discussed for the last 75-odd years [2, 4, 12, and the references there], of the possible existence of The X-ray phase analysis of the specimens was per- stable low-temperature cubic C-modifications in Ce formed on an HZG-4 diffractometer (Carl Zeiss, Jena, sesquioxides reversibly transforming into A- or B-mod- Germany), CuKα radiation, Ni-filter, and Si external ifications during heating. It was shown [14] that, if the standard. The initial specimen was preliminarily

282 FEDOROV et al.

(a) G T, °C T, °C 2400 X H 751

666 2000 DTG

DTA 512 1600 ABC 422 341 1200 292

Fig. 2. Derivatogram of thermal decomposition of hydrated 800 lanthanum oxide.

ground in a jasper mortar. The angular range was 10°– 60°; the recording rate was 2°/min. 400 Pr2O3 Pm2O3 Eu2O3 Tb2O3 Ho2O3 Tm2O3 Lu2O3 The positions and intensities of the diffraction peaks

La2O3 Nd2O3 Sm2O3 Gd2O3 Dy2O3 Er2O3 Yb2O3 were calculated by the Profit and Powder2 programs with allowance for both CuKα and CuKα lines. Sub- 1 2 0 57 59 61 63 65 67 69 71 stance identification was done by comparing the experimental data with the data of the JCPDS Powder Data Base (1997). The indexing of diffraction patterns and calculation of the lattice parameters were done by the (b) 2500 LS method within the Powder 2 program. Melt The X-ray diffraction patterns obtained from the X H starting materials were essentially different from those of La2O3 and Nd2O3. Their comparison with the Data Base 2000 data showed that these lines corresponded to La(OH)3 and Nd(OH)3, sp. gr. P63/m (JCPDS no. 36-1481, a = 6.528 Å and c = 3.858 Å and no. 06-0601, a = 6.421 Å and c = 3.74 Å, respectively). 1500 ABC As is seen from Fig. 2, the decomposition of hydrated lanthanum oxide proceeds in three stages. The weight losses correspond to the following reactions: at ° 1000 the first stage at T = 292 C, 2La(OH)3 La2O3 á H2O + 2H2O and at the second and third stages (T = 422°C), 500 La2O3 á H2O > La2O3 + H2O. La Ce Pr Nd Pm Sm Eu Cd Tb Dy Ho Er TmYbLu The above temperatures correspond to the beginning of 1.05 1.00 0.95 0.90 0.85 the decomposition processes on the curve of the weight ri, Å loss (G). These equations were confirmed by the diffraction patterns of the samples heated to certain temperatures corresponding to the processes that were Fig. 1. Phase transition in the R2O3 series: (a) according to fixed on derivatograms (Fig. 3). DTA data [6]: ᭪ solidification points, × transformation points; X-ray phase analysis data [1]: ᭝ transformation point dur- The X-ray diffraction pattern of La2O3 á H2O ing temperature increase, ᭞ transformation point during (Fig. 3b) is similar to that of LaO(OH): monoclinic sys- lowering of the temperature according to data [1]; Ճ trans- tem, sp. gr. P2 /m, a = 4.417 Å, b = 3.929 Å, c = 6.572 Å, Ճ 1 formation points according to data [13]; transformation β = 112.5° [15] (JCPDS no. 19-0656). However the lat- points of Pm2O3 as functions of the nucleus charge of an N-cation; (b) according to data [4] with no allowance for the tice parameters of La2O3 á H2O are essentially different metastable states dependent on the ionic radius R3+. from those of LaO(OH) (according to our data, a =

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 ON POLYMORPHISM AND MORPHOTROPISM 283

I/I0

(a) La(OH)3

(b) La2O3 á H2O

(d) La2O3

20 30 40 50 2θ, deg

Fig. 3. X-ray diffraction patterns from dehydrated lanthanum oxide at different stages of dehydration; (a) initial sample; heating to (b) 360, (c) 600, and (d) 1000°C.

4.436 Å, b = 3.930 Å, c = 6.619 Å, β = 112.12°).The ond and third stages of decomposition, respectively. diffraction pattern taken upon the second stage of The Data Base indicates for La2O3 (JCPDS no. 05-0602) decomposition (Fig. 3c) was almost identical to that of completely dehydrated oxide. The unit-cell parameters the sp. gr. P3m1 , a = 3.937 Å, c = 6.129 Å. The anal- of the trigonal lattice were determined as a = 3.923 Å, ysis of the additional weak reﬂections on the diffraction c = 6.124 Å and a = 3.935 Å, c = 6.127 Å upon the sec- pattern observed upon the second stage of decomposi-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 284 FEDOROV et al.

G T, K T, °C Gd Eu A 2000 Sm Pm 1500 B 760 DTG 1000

DTA 590 500

492 0.93 0.94 0.95 0.96 0.97 r, Å 390 312 Fig. 5. Approximation of the data on the temperatures of 225 A B phase transitions [6, 13]. Solid line describes the equation T = 3607(0.977 – r)0.1323, dash line (r = 0.977 Å) Fig. 4. Derivatogram of thermal decomposition of hydrated is the boundary of the morphotropic transformation. The neodymium oxide. ionic radii were taken at c.n. 6 [19]. tion and disappearing upon the third stage (Fig. 3c) tern from LaO(éH). It was indexed in the monoclinic indicates the presence of several percent of lanthanum system a = 4.388 Å, b = 3.870 Å, c = 6.392 Å, β = oxicarbonate La2O2CO3 (JCPDS no. 37-0804). These 112.2° (Table 1), which is consistent with [16], where reﬂection disappeared upon the third stage of decompo- it was stated that all the RE oxyhydroxides are isostruc- sition (Fig. 3d). tural. The Data Base provides no indexing of these pat- The decomposition of Nd(OH)3 (Fig. 4) proceeds in terns (JCPDS no. 13-0167). three stages. The calculation of the weight losses corresponds to the following reactions at T = 225, 390, 590°C: Upon the second phase of decomposition, the C-phase of neodymium oxide was detected (cubic face centered 2Nd(OH)3 Nd2O3 á H2O + 2H2O, lattice, a = 11.078 Å), which, in actual fact, is a hydrate Nd2O3 á H2O Nd2O3 á 0.3H2O + 0.7H2O, of the composition Nd2O3 á 0.3H2O. In the temperature ° Nd O á 0.3H O Nd O + 0.3H O. range 590Ð760 C, the Nd hydrate, Nd2O3 á 0.3H2O, 2 3 2 2 3 2 decomposes to the Nd O of the A type corresponding The X-ray diffraction pattern of the sample upon the 2 3 ﬁrst stage of water loss corresponded to the formula to JCPDS no. 06-0408 (P3m1 , trigonal system, a = NdO(éH) and was similar to the X-ray diffraction pat- 3.831 Å and c = 5.999 Å).

Table 1. Indexing of X-ray diffraction pattern of NdOOH θ, 4 2 4 2 ∆ 4 2 2 deg d (obs), Å 10 /d (obs) I/I0 h k l 10 /d (calc) 10 /d 14.900 5.9409 283.3 17 0 0 1 285.6 Ð2.30 21.320 4.1642 576.7 42 Ð1 0 1 577.0 Ð0.32 21.709 4.0905 597.6 52 1 0 0 606.1 Ð8.51 27.679 3.2203 964.3 100 0 1 1 953.3 11.01 29.871 2.9888 1119.5 47 Ð1 0 2 1119.1 0.35 31.089 2.8744 1210.3 18 1 0 1 1206.6 3.75 31.918 2.8016 1274.1 58 Ð1 1 1 1259.1 15.00 38.339 2.3459 1817.1 26 0 1 2 1810.2 6.93 41.165 2.1911 2082.9 20 Ð2 0 1 2080.7 2.24 43.424 2.0822 2306.5 14 Ð2 0 2 2308.1 Ð1.50 45.944 1.9737 2567.1 18 0 0 3 2570.7 Ð3.62

47.372 1.9175 Nd2O3 20 Nd2O3 Nd2O3 Nd2O3 50.742 1.7978 3093.9 22 2 1 0 3092.3 1.67 57.285 1.6070 3872.3 22 1 2 1 3877.2 Ð4.90

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 ON POLYMORPHISM AND MORPHOTROPISM 285

| |n Table 2. Parameters of the equation T = ‡ r0 – r which describe the upper temperature boundary of stability of various families of phases Correlation Compound r , Å a, K/Å n Reference 0 coefﬁcient

Pb4R3F17 1.230* 1430 0.091 0.9989 20 Ca8R5F31 1.165* 1800 0.136 0.9994 20 Ca2RF7 1.171* 2030 0.192 0.9997 20 β ± RF3( -YF3) 1.226 0.005* 3160 0.284 0.960 15 ± ± ± B-R2O3 0.977 0.014** 3600 950 0.132 0.093 0.9893 Present study * Ionic radii of ﬂuorides at c.n. 8 [19]. ** Ionic radii of oxygen at c.n. 6 [19].

DISCUSSION OF RESULTS at c.n. 6. The approximation obtained is characterized The results obtained in our study confirm and sup- by a high value of the correlation coefficient. plement the X-ray diffraction data [4, 5] on the transi- Table 2 lists the parameters of this equation, which tion of the cubic C-modification of neodymium oxide describe the stability boundaries of various phases. The to the trigonal A-modification accompanied by the deviation of n from the value 0.25 can be explained by removal of volatile impurities. Therefore, this transfor- the fact that the transition temperatures are rather far mation cannot be considered as a polymorphic transi- from the temperature of absolute zero. The value r0 = tion proper. The stages and the hydration temperatures 0.977 Å lying between the ionic radii of Pm and Nd sets observed in this study agree with the data in [4, 5, 16, 17]. the boundary of the morphotropic A B transforma- The inconsistencies can be explained by the different tion in R O oxides. The data on the B C polymor- methods used for preparing the initial hydrated forms. 2 3 phic transitions for R2O3 have not been processed quan- According to our data, the dehydration of neodymium titatively because of the insufficient number of points. oxide is completed at a rather high temperature, 760°C. It seems that the presence of strongly bound (residual) In [7], some additions to the scheme of polymor- water explains the error made by Warshaw and Roy, phism and morphotropism in R2O3 series were made which concerned high-temperature polymorphism in who described the reversible transitions in the R2O3 oxides of the cerium group in the presence of H O [1]. yttrium oxides (the existence of the A- and B-modifica- 2 tions in the narrow temperature intervals). The conclu- They considered water only as a means for accelerating sions drawn are based on the data for samples contain- the phase transition but not affecting the chemical com- ing 10 mol % of heterovalent impurities (Sr and Ca position (i.e., as a catalyst). However, in actual fact, a oxides but not Mg and Ba oxides). We believe that such small amount of residual water in the substance an approach is erroneous because it ignores the possible changes the pattern of polymorphism and morphotro- stabilization of unstable crystalline modifications by pism. impurities [4] accompanied by the formation of indi- The structural changes occurring in R2O3 in the tran- vidual berthollide phases separated from the ordinate of sition from the C- to the A-modification can be the corresponding component [15, 21]. described by the mechanism of crystallographic shear [5, 10, 11, 18]. Thus, of two alternative schemes of polymorphism ACKNOWLEDGMENTS and morphotropism in the series of RE sesquioxides, The authors are grateful to Kh.S. Bagdasarov, the scheme in Fig. 1b seems to be more adequate. Pro- Yu.V. Pisarevskiœ, and F.M. Spiridonov for their interest ceeding from the changes in the melting points in the in this study and discussion of the results. series of RE oxides with due regard for the “gadolinium kink,” the whole series can be divided into two groups—cerium and yttrium. The model of an ideal REFERENCES morphotropic series can be applied to both of these 1. J. Warshaw and R. Roy, J. Phys. Chem. 65, 2048 (1961). groups [14]. This situation corresponds to RE fluorides 2. G. Brauer and R. Muller, Z. Anorg. Allg. Chem. 321 (5Ð6), [15]. Figure 5 shows the results of the LS procedure 234 (1963). applied to data [6, 13] on the temperatures of the A B 3. M. Foex and J.-P. Traverse, C. R. Seances Acad. Sci. 262, polymorphic transformations in accordance with the 636 (1966). | |n limiting thermodynamic equation T = a r0 – r , where 4. V. B. Glushkova, Polymorphism of Rare-Earth Oxides ± ± ± a = 3600 950, r0 = (0.977 0.014) Å, and n = 0.132 (Nauka, Leningrad, 1967). 0.093. The ionic radii (tabulated to the third decimal 5. V. S. Rudenko and A. G. Bogdanov, Izv. Akad. Nauk place) were taken from the system suggested in [19] SSSR, Neorg. Mater. 6 (12), 2158 (1970).

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6. J.-P. Traverse, Contribution au developpement de méth- 13. F. Weigel and U. Scherer, Radiochim. Acta 4, 197 odes d’experimentation a temperature elevée. Applica- (1965). tion a l’étude du polymorphisme des sesquioxydes de 14. P. P. Fedorov, Kristallografiya 40, 308 (1995) [Crystal- terres rares et des changements de phases dans les sys- logr. Rep. 40, 278 (1995)]. tems zirconeÐchaux et zirconeÐoxyde de strontium, 15. P. P. Fedorov and B. P. Sobolev, Kristallografiya 40, 315 Thése (L’Univérsite Scientifique et Medicale de Greno- (1995). ble, Grenoble, 1971). 16. V. P. Klevtsov and L. P. Sheina, Izv. Akad. Nauk SSSR, 7. L. M. Lopato, F. V. Shevchenko, A. E. Kushchevskiœ, and Neorg. Mater. 1 (12), 2219 (1965). S. G. Tresvyatskiœ, Izv. Akad. Nauk SSSR, Neorg. Mater. 10 (8), 1481 (1974). 17. B. N. Rybakov, A. F. Moskvicheva, and G. D. Bere- govaya, Zh. Neorg. Khim. 14 (11), 2904 (1969). 8. P. Duran, Bull. Soc. Fr. Ceram., No. 102, 47 (1974). 18. B. G. Hyde, Acta Crystallogr., Sect. A: Cryst. Phys., 9. J. Coutures, A. Rouanet, R. Verges, and M. Foex, J. Solid Diffr., Theor. Gen. Crystallogr. A27 (6), 617 (1971). State Chem. 17 (1/2), 171 (1976). 19. R. D. Shannon, Acta Crystallogr., Sect. A: Cryst. Phys., 10. O. Greis, J. Solid State Chem. 34 (1), 39 (1980). Diffr., Theor. Gen. Crystallogr. A32 (5), 751 (1976). 11. C. Boulesteix, Handbook on the Physics and Chemistry 20. P. P. Fedorov, L. V. Medvedeva, and B. P. Sobolev, Zh. of Rare Earth, Ed. by K. A. Gschneidner and L. Eyring Fiz. Khim. 67 (5), 1073 (1993). (North-Holland, Amsterdam, 1982), Chap. 44, p. 321. 21. P. P. Fedorov, P. I. Fedorov, and B. P. Sobolev, Zh. Neorg. 12. B. I. Pokrovskiœ and F. M. Spiridonov, in Compounds of Khim. 18 (12), 3319 (1973). Rare-Earth Elements. Systems with Oxides of Elements from Groups IÐIII, Ed. by I. V. Tananaev (Nauka, Mos- cow, 1983), p. 17. Translated by L. Man

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 287Ð290. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 322Ð325. Original Russian Text Copyright © 2002 by Zakharchenko, Timonin, Sviridov, Aleshin.

PHASE TRANSITIONS

Structural Phase Transitions in Heteroepitaxial Ferroelectric Films I. N. Zakharchenko, P. N. Timonin, E. V. Sviridov, and V. A. Aleshin Research Institute of Physics, Rostov State University, pr. Stachki 8, Rostov-on-Don, 344090 Russia Received October 18, 1999; in ﬁnal form, March 7, 2001

Abstract—The changes in the structural–deformational characteristics occurring during the phase transitions in the crystal lattice of epitaxially grown thin (Ba,Sr)TiO3 ﬁlms on the (001) cleavage of MgO crystals have been studied. It is shown that microdeformations along the surface normal of the wall of a c-domain ﬁlm increase with the temperature and attain maximum values in the vicinity of the phase transition. No decrease in the dimensions of the coherent-scattering regions was observed. The comparison of the deformation of the “average” lattice and microdeformations led to the assumption that the transition from the paraelectric to the ferroelectric phase is accompanied by the dislocation-induced formation of a dipole-glass-type intermediate phase at the temperatures exceeding that of the phase transition. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION but becomes nonpolar—the etching figures of domains observed in an electron microscope disappeared [3, 4]. In recent decades, intensive studies of the physical These anomalies were explained by the effect of ther- properties of thin ferroelectric films and phase transi- moelastic stresses and the anisotropic distribution tions occurring in these films have been performed. The of point defects in the films associated with the tech- properties of these films often differ from the properties nology of film formation (high-frequency cathode sput- of bulk specimens of the same compositions: in partic- tering). ular, phase transitions in the films are often diffuse and are similar to those observed in ferroelectricsÐrelaxors The above results allow one to assume that the [1, 2]. restructuring of the crystal lattice of the films during phase transitions is characteristic of both ferroelectrics Studying thin epitaxial (Ba,Sr)TiO3 films grown on and ferroelectrics–relaxors and, thus, should reflect the the (001) surface of MgO crystals, we established that elastic interaction between the film and the substrate. the degree of phase-transition diffusion depends on the The diffusion of phase transitions in ferroelectricsÐ perfection of the crystal structure characterized by the relaxors is associated with the formation of an interme- dimension of the coherent-scattering region, D, and the diate dipole-glass phase prior to the transition from the ε value of the average microdeformation ˜ 3 = paraelectric to the ferroelectric phase. Crystal restruc- 2 turing is studied in most detail for lead magnesium nio- 〈〉()∆d/d along the [001] direction (where d is the bate and its solid solution with lead titanate [5, 6]. The ε corresponding interplanar spacing) [3]. The ˜ 3 values specific features of diffuse scattering and the changes in of various films at room temperature are about 10–3–10–4, the widths and the intensities of Bragg reflections are whereas the D values exceed 1000 Å; the degree of interpreted as the result of the formation of mesoscopic phase-transition diffusion is higher for films with polar clusters in the nonpolar cubic matrix whose ε˜ dimensions increase with lowering of the temperature. higher 3 values [3], i.e., depends on the presence of In turn, macroscopic regions with nonuniform sponta- linear defects (dislocations). It is also shown that the neous deformation provide the formation of a dipole- ferroelectric phase transition occurs at temperatures glass phase [5]. By measuring the broadening of the higher than in the bulk specimens of the same compo- Bragg reflections, we determined the average microde- sitions because of the compressive thermoelectric formation of the crystal lattice along the directions cor- stresses caused by the difference in the thermal expan- responding to the reflection indices and, thus, evaluated sion coefficients of the film and the substrate [4]. the degree of structure disorder in both dipole-glass and In some instances, the values of the lattice parame- other phases. ters c were higher and the values of the a parameters, The first-order phase transition in the (Ba,Sr)TiO3 lower than the corresponding values for the bulk speci- films [7] is accompanied by the nucleation of a new men. Also, at temperatures T exceeding the phase-tran- phase, with both phases being coexistent within a cer- sition temperature TC, the unit cell remains tetragonal tain temperature interval [8]. As a result, the dimen-

288 ZAKHARCHENKO et al.

ε × 3 a, c, Å 3 10 EXPERIMENTAL 4.02 1.2 The (Ba0.7, Sr0.3)TiO3 films were grown by high-frequency sputtering of polycrystal line targets of stoichi- 1 ometric composition [15] in an oxygen atmosphere under a pressure of 93 Pa. The films were deposited 4.00 1.0 onto as-cleaved MgO (001) surfaces. The substrates were preliminarily heated up to 450°C by a resistive heater and then bombarded with the plasma particles of 2 a high-frequency discharge, which heated the substrate 3.98 0.8 up to 850°C. The power of the high-frequency discharge was 25 W/cm2, the deposition rate was 130 Å/min. Upon the attainment of the desired film thickness (0.5– 2.0 µm), the high-frequency discharge was switched 3 ° 3.96 0.6 off, and the film was spontaneously cooled to ~250 C. The high heat capacity of the heater provided sufficient inertial cooling at an average rate of 5°C/min. In the temperature range from 250°C to room temperature, the average cooling rate was maintained at a level of 3.94 0.4 ° 0 100 200 300 400 500 600 about 1 C/min, which resulted in the formation of the equilibrium domain structure in the film. T, °C The X-ray diffraction studies were made on a Fig. 1. Temperature dependence of the parameters of the DRON-3 diffractometer (filtered CuKα radiation). The average unit cell: (1) c(T), (2) a(T), and (3) the average specimens were placed into a special chamber fixed on ε˜ microdeformation along the [001] direction, 3(T). a goniometer, which allowed one to record the reflections within the temperature range 20Ð550°C. The phase transitions were recorded during the cooling of sions of the blocks of the crystal lattice decreased [9]. ° ε˜ Kenzig [10] observed the broadening of reflections the specimen preliminarily heated up to 500 C. The which could be associated with a decrease in the dimen- and D values were obtained by the approximation sions of coherent-scattering regions. method [6]. The reflection profiles were approximated by Gaussians. The present study is dedicated to structural phase transitions in heteroepitaxial ferroelectric films. We studied the temperature dependence of the unit-cell RESULTS AND DISCUSSION parameters and the changes in the structuralÐdeforma- According to the X-ray diffraction data, the (001) tional characteristics, such as the average microdefor- planes and the [100] and [010] directions in the films mation and the dimension of the coherent-scattering and in the substrates were parallel. The c-axis of the regions during phase transitions. The temperature film was normal to the surface (the “c-domain dependence of the “average” unit-cell deformation [11] films”). The method used in our study is described was compared with the temperature dependence of the elsewhere [17]. average microdeformation during phase transitions. The changes in the structuralÐdeformational charac- It would be interesting to compare the results of the teristics during phase transitions were studied on the films with a rather high degree of structural perfection. traditional studies of the temperature dependence of the dielectric constant obtained earlier [12Ð14] (not consid- Figure 1 (curves 1 and 2) shows the temperature ered here) with the structural data. It was shown both dependence of the lattice parameters for a 0.6-µm-thick experimentally and theoretically (within the frame- film, whose average microdeformation along the direc- ε × –4 works of the thermodynamic approach) that the biaxial tion normal to the film surface was 3 = 7.7 10 at D > compressive stresses usually observed in epitaxial 1000 Å at room temperature. As was indicated earlier, the kink in the temperature curve of the c- and a-param- (Ba,Sr)TiO3/MgO films suppress the formation of the maximum for the ε component of the dielectric-con- eters, indicating the occurrence of the phase transition, 11 is observed at a considerably higher temperature than stant tensor in the temperature range corresponding to the phase-transition temperature in the bulk specimen the transition from the tetragonal to the cubic phase. In (35°C [18]). At T > T , the unit-cell symmetry ε ε C this case, the main components 11 = 22 of the dielec- remained tetragonal. The c-parameter was determined tric-constant tensor readily measured in epitaxial films using the 004 reflection, whereas the a-parameter was on flat dielectric substrates provide no additional infor- determined from the angular positions of the 004 and mation. 224 reflections.

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STRUCTURAL PHASE TRANSITIONS 289

No decrease in the dimensions of the coherent-scat- ε , ε~ tering regions along the [001] direction (characteristic t t of ferroelectric films undergoing the first-order phase 0.004 transition) was observed in the whole temperature β β 0.003 range studied. The broadening ratio 1/ 2 for the 00l1 and 00l2 reflections was equal to the tangent ratio of the corresponding Bragg angles: in other words, reflection 0.002 broadening was caused by microdeformations alone. Figure 1 (curve 3) shows the temperature depen- 0.001 1 dence of ε˜ whose squared value is, in fact, the S 3 33 0 component of the tensor of averaged random microde- 2 formations S = 〈ε ε 〉 with i, j = 1, 2, …, 6 [19]. The T3 T2 T1 ij i j –0.001 maximum (S )1/2 = ε ~ 10–3 values were observed in ~ 33 3 ε ε the vicinity of the phase transition. v v 0.004 The range of the angular block misorientation with 4 0 respect to the film normal determined by the 1θ-scanning at a narrow slit of the static counter remained constant. 0.003 Analyzing the data obtained, we compared the tem- –0.005 ε perature dependences of uniform volume ( v) and shear 0.002 ε ( t) deformations of the average unit cell with the tem- –0.010 perature dependences of the average volume (ε˜ ) and v 0.001 3 ε ε ε –0.015 shear (˜ t ) microdeformations. The v and t values were determined from the following equations: T3 T2 T1 0 –0.020 ε 〈〉ε 〈〉 ε 0 100 200 300 400 500 600 v = 2 1 + 3 , T, °C ε 〈〉ε 〈〉 ε t = 3 Ð 1 , where Fig. 2. Temperature dependence of the deformation of the average unit cell and microdeformation: (1) average shear 〈〉ε = ()a Ð a /a , ε˜ 1 T 0 0 microdeformation t , (2) shear deformation of the average 〈〉ε () unit cell ε , (3) average volume microdeformation ε˜ , and 3 = aT Ð a0 /a0, t v (4) volume deformation of the average unit cell εv. and aT and cT are the average unit-cell parameters at the 〈ε 〉 〈ε 〉 temperature T. The 1 and 3 deformations were calculated with respect to the unit-cell parameters a0 and Figure 2 illustrates these dependences. During cool- c0 measured at the maximum temperature from which ing, the first kink on the ε˜ (T) plot (curve 1) is observed the films were cooled. t at T ≈ 110°C. An increase in ε recorded at this temper- ε ε 1 t The ˜ v and ˜ t values were determined from the ature is accompanied by a drastic decrease in the uni- ε equations relating reflection broadening and the com- form shear deformations t in the averaged unit cell ponents of the tensor of averaged random microdefor- (curve 2). In a ferroelectric undergoing a phase transi- mation, Sij [19]. Since the tetragonality of the unit cell tion, the dependence of the c- and a-parameters at fur- ε in the temperature range studied was low (c/a < 1.01), ther cooling provides an increase in t [8]. An increase ε ≈ ε we used the well-known relationships for the cubic in t on curve 2 begins only at T < T2 91°ë upon the t crystal S11 = S22 = S33 and S12 = S13 = S23. Then, reflection jump at T = T2 (where T2 corresponds to the kink on the broadening during θ/2θ scanning along the scattering curves c(T) and a(T) (Fig. 1, curves 1, 2)). At tempera- vector is given by the equations tures lower than T2, the behavior of the shear microde- 2 2 ε β θ formations is changed and ˜ t starts decreasing. Finally, 00l = 16S11 tan 00l, at T < T ≈ 63°ë, both ε and ε˜ become temperature- β2 ()θ2 3 t t ll2l = 16/3 S11 + S12 tan ll2l, independent. whence, using the measured β and β values and 004 224 ε ε The t and ˜ t dependences observed lead to the the calculated S11 and S12 values, we obtain assumption that, in the temperature range T2 < T < T1, ε 〈〉()ε ε ε 2 () ˜ v ==1 ++2 3 3 S11 + 2S12 , an intermediate dislocation-induced dipole-glass phase ε should exist [20]. Indeed, an increase in ˜ t at T1 indi- ε˜ 〈〉()ε ε 2 () t ==1 Ð 3 2 S11 Ð S12 . cates the appearance of spontaneous random displace-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 290 ZAKHARCHENKO et al.

ε ε REFERENCES ments giving rise to lattice disordering because ˜ t > t. At T2, the averaged lattice becomes polar, and the shear 1. Z. Surovyak, A. M. Margolin, I. N. Zakharchenko, et al., deformations of the average unit cell drastically Thin Solid Films 176, 227 (1989). increase and start exceeding ε˜ , which also indicates 2. I. N. Zakharchenko, E. V. Sviridov, V. A. Alyoshin, and t L. A. Sapozhnikov, in Abstracts of the Second Interna- the formation of macroscopic regions of rather uniform tional Seminar in Relaxor Ferroelectrics, Dubna, 1998, polarization. At T3 < T < T2, a mixed phase is observed p. 112. in which the average polar lattice coexists with the 3. Z. Surovyak, A. E. Panich, and V. P. Dudkevich, Thin regions of the dipole-glass phase in which the direc- Ferroelectric Films (Rostovskiœ Pedagog. Univ., Rostov- tions of local dipole moments are disordered. At T < T3, on-Don, 1994). the random shear microdeformations ε˜ decrease to the 4. V. M. Mukhortov, Yu. I. Golovko, V. A. Aleshin, et al., t Phys. Status Solidi A 77 (1), K37 (1983). values approximately equal to those observed in the paraelectric phase at T T , whereas ε considerably 5. O. A. Bunina, I. N. Zakharchenko, P. N. Timonin, and > 1 t V. P. Sakhnenko, Kristallografiya 40 (4), 708 (1995) ε exceeds ˜ t . Thus, below T3, only the macroscopically [Crystallogr. Rep. 40, 655 (1995)]. homogeneous ferroelectric phase exists. 6. S. B. Vakhrushev, Author’s Abstract of Doctoral Disser- tation in Physics and Mathematics (St. Petersburg, It should be emphasized that the dependences of the 1998). volume microdeformations also have anomalies at tem- 7. L. A. Shebanov, Uch. Zap. Latv. Univ. im. Petra Stuchki peratures T1, T2, and T3 (curve 3 in Fig. 2), which is con- 189, 150 (1974). sistent with the conclusion regarding the formation of 8. G. A. Smolenskiœ, V. A. Bokov, V. A. Isupov, et al., The intermediate phases. At the same time, the dependence Physics of Ferroelectric Phenomena, Ed. by G. A. Smo- of the volume deformation of the average unit cell cor- lenskiœ (Nauka, Leningrad, 1985). responds to the behavior of a conventional ferroelectric 9. Yu. V. Zvirgzde and Ya. Ya. Kruchan, Izv. Akad. Nauk crystal (Fig. 2, curve 4). Latv. SSR, Ser. Fiz. Tekh. Nauk, No. 5, 57 (1976). It seems that the elastic interaction between the film 10. W. Kenzig, Helv. Phys. Acta 24 (2), 175 (1951). and the substrate increases the phase-transition temper- 11. M. A. Krivoglaz, Theory of X-ray and Thermal Neutron ature and causes the tetragonal distortion of the cubic Scattering by Real Crystals (Nauka, Moscow, 1967; Ple- num, New York, 1969). unit cell in the paraelectric phase. 12. I. N. Zakharchenko, Ya. S. Nikitin, V. M. Mukhortov, et al., Phys. Status Solidi A 114, 559 (1989). 13. S. Desu, Z. J. Chen, P. V. Dudkevich, et al., Mater. Res. CONCLUSION Soc. Symp. Proc. 401, 195 (1996). In the films studied, the ferroelectric tetragonal 14. Z. Surowiak, V. M. Mukhortov, and V. P. Dudkevich, phase is formed in the tetragonal and not in the cubic Ferroelectrics 133, 1 (1993). 15. V. M. Mukhortov, Yu. I. Golovko, Vl. M. Mukhortov, matrix. At T > TC, dislocations break the lattice periodicity and give rise to microdeformations; they are also et al., Pis’ma Zh. Tekh. Fiz. 5, 1175 (1979) [Sov. Tech. sources of fluctuations in the local electric fields. This, Phys. Lett. 5, 492 (1979)]. in turn, leads to the diffusion of the phase transition and 16. V. I. Iveronova and G. P. Revkevich, Theory of X-ray the formation of intermediate phases during such trans- Scattering (Mosk. Gos. Univ., Moscow, 1972). formations. During the restructuring of the crystal lat- 17. M. G. Radchenko, E. V. Sviridov, E. I. Bondarenko, et al., Available from VINITI No. 6268-84 (Rostovs. tice, some blocks that have undergone phase transitions Univ., Rostov-on-Don, 1984). at a higher temperature do not form individual coher- 18. V. A. Bokov, Zh. Tekh. Fiz. 27 (8), 1784 (1957) [Sov. ent-scattering regions and can promote only higher Phys. Tech. Phys. 2, 1657 (1958)]. microdeformations. 19. P. N. Timonin, I. N. Zakharchenko, O. A. Bunina, and V. P. Sakhnenko, Phys. Rev. B 58 (6), 3015 (1998). 20. P. N. Timonin, Zh. Éksp. Teor. Fiz. 116 (3), 986 (1999) ACKNOWLEDGMENTS [JETP 89, 525 (1999)]. This study was supported by the Russian Founda- tion for Basic Research, project no. 98-02-18069. Translated by L. Man

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 291Ð297. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 326Ð332. Original Russian Text Copyright © 2002 by Karpenko, Kyarov, Temrokov, Vinokurskiœ. PHASE TRANSITIONS

Polymorphous Transformations of the B1ÐB2 Type in Crystals of Finite Dimensions S. V. Karpenko, A. Kh. Kyarov, A. I. Temrokov, and D. L. Vinokurskiœ Research Institute of Applied Mathematics and Automation, Kabardino-Balkar Research Center, ul. Shortanova 89a, Nalchik, 260000 Russia Kabordino-Balkar State University, ul. Chernyshevskogo 142, Nalchik, 360004 Russia Received January 10, 2001; in ﬁnal form, May 16, 2001

Abstract—High-pressure-induced structural phase transitions in alkali halide crystals have been studied. The pressure of the B1ÐB2 phase transition in crystals of finite dimensions is studied with the use of pair-interaction potentials within the theory of inhomogeneous electron gas. It is shown that the pressure of polymorphous transformation depends on the initial crystal dimensions. The anomalous dependence of the transition pressure on the dimensions of crystalline grains in lithium fluoride is considered and interpreted. Some structural and surface characteristics of the B1 and B2 modifications of a number of alkali halide crystals are calculated in the approximation of seven coordination spheres. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION chlorides. The best agreement with the experimental data was attained for the cohesion energy of the The phase transition between the B1-type (sodium B1 phase and under the transition pressure. Later chloride structure) and the B2 (cesium chloride struc- Cohen and Gordon [9] studied these phases in more ture) phases of alkali halide crystals is being studied detail—the short-range potentials also took into both experimentally and theoretically. The experimen- account the quantum corrections for inhomogeneity of tal study is performed by two main methods—speci- kinetic energy; they also made an attempt to consider men compression by a shock wave and the creation of the effect of the second and the third nearest neighbors. static pressure in a diamond cell. Bridgman [1] used the However, the allowance for the contributions from the method of static compression and observed the phase ions of the second and third coordination spheres often transitions in KCl and KBr crystals in the pressure deteriorated the agreement between the theory and the range 20Ð30 kbar. Later, Al’tshler [2, 3] studied shock- experiment. wave induced compression in some alkali halide crystals. His experimental results confirmed the Bridgman It should be noted that all the above investigations data for potassium chloride and bromide, and thus were made for infinite crystals. In [10], the effect of the proved the possibility of a polymorphous transforma- surface contribution to the thermodynamic potential of tion in these compounds in the vicinity of 20 kbar. The a crystal on the properties of the B1ÐB2 phase transition first detailed theoretical study of the B1ÐB2 phase tran- was studied for a number of alkali halide crystals. sition was made by Jacobs [4] using the short-range BornÐMayer potential with due regard for the interac- In recent years, the development of the high-pres- tions only between the nearest neighbors [5]. Despite sure methods triggered studies of the properties of the fact that the pressures of polymorphous transforma- polymorphous transformations in crystalline sub- tions obtained in [5] only approximately corresponded stances [11Ð15]. However, despite the considerable to the experimental data, Jacobs determined some progress in the experimental and theoretical studies of important characteristics of this transition: the change polymorphous transformations, many problems still ∆ in the crystal volume Vt under the pressure of the remain to be solved. Thus, while the structure and prop- phase transition and the jump in the lattice constant. erties of high-pressure phases have been insufficiently Born and Huang [6] and Tosi [7] used the refined Born– studied, the polymorphous transformations in crystals Mayer approximation to determine the phase transition of finite dimensions have been hardly studied at all. pressure and the cohesion energy of the B1 and B2 The present investigation is dedicated to the charac- phases. teristics of polymorphous transformations in ionic crys- Later, Gordon and Kim used the model of an inho- tals of finite dimensions. Special consideration is given mogeneous electron gas [8] to calculate the cohesion to the calculation of the elastic characteristics of crys- energy of the B1 and B2 phases and the pressure of the tals and their dependence on the pressure applied. The polymorphous transformations in the crystals of lith- knowledge of these values is necessary for understand- ium, potassium, and sodium bromides, fluorides, and ing the nature of the binding forces in ionic crystals. All

292 KARPENKO et al. the calculations were performed in the approximation where σ(hkl) is the surface energy of the (hkl) face, of the zero absolute temperature. ()i W j is the energy of a particle in the jth layer provided ()i by the ith type of the interacting forces, W∞ is the SIZE EFFECT IN THE B1ÐB2 TRANSITION same in the crystal bulk, and nj(hkl) is the number of IN SMALL CRYSTALS particles in the jth plane per unit area. In the zeroth Proceeding to polymorphous phase transitions in approximation used, Eq. (2) acquires the form [19] alkali halide crystals of finite dimensions, we have to modify the thermodynamic potential of the crystal ()i ()i σ()hkl = n ()hkl ∑()W Ð W∞ . (3) under the conditions of hydrostatic pressure [16] by 0 0 adding a new term, which describes the surface energy i of the crystal. To simplify the model, we assume that Now, consider a plane net inside an infinite solid. Obvi- the surface of a crystal of the B1 phase is (100)-faceted, ously, for a perfect (undistorted) crystal, we have whereas a crystal of the B2 phase is (110)-faceted, ()i ()i ()i because, under zero external pressure and absolute-zero W∞ = WS + 2W ν/2 , (4) temperature, these faces possess minimum surface ()i energies. Thus, the thermodynamic potential of such a where WS is the energy of a particle on the net pro- crystal, with due regard for the surface contribution, vided by the ith type of interaction with all the remain- can be written in the form ()i ing particles in this plane, and W ν/2 is the energy of the 7 same particle determined by its interaction with all the ()i ()i ()()i ()i GB1 = ∑ Nk Uk ak R other particles in the planes above and below the plane k = 1 under consideration. Thus, the particle energy in the (1) 7 surface plane of an undistorted crystal is () ∂ () () () () () () i i i i i i i 2 ()i ()i ()i Ð V ------∑ N U ()a R Ð4αµ /R + πr kσ, ∂ ()i k k k W0 = WS + W ν/2 . (5) V k = 1 ()i Eliminating W ν/2 from Eqs. (4) and (5) and substitut- where αµ = 1.747558 and αµ = 1.76268 are the Made- 1 2 ing the result into Eq. (3), we obtain lung constants of the B1 and B2 structures, R1 and R2 are the distances between the nearest neighbors in these ()i ()i 3 3 σ()()()() β hkl = 1/2 n0 hkl ∑ Ð 1 W∞ , (6) modifications, V1 = 2R1 and V2 = (8/33 )R2 are the i unit volumes of the B1 and B2 phases, UB1(R1) and () () UB2(R2) are the pair-interaction potentials, ak = Rk/R0 is W i A i the ratio of the radius of the kth coordination sphere to β(i) S S where = ------()i = ------()i is the ratio of the sums over the the radius of the first one, and Nk is the coordination W∞ Aν number. The subscript i enumerates both B1 and B2 infinite plane net to the sum over the infinite lattice for phases. It should be noted that we used numerical val- the ith type of the interionic forces. In particular, for ues of the pair potentials obtained by the self-consistent Coulomb forces, β is the ratio of the Madelung constant method [17] within the theory of inhomogeneous for the plane net to the Madelung constant for the three- electron gas, with r being the initial radius of the crys- dimensional lattice. For other forces, β is the ratio of the tal, σ, the surface energy, and k, the numerical coeffi- rapidly converging series whose summation presents cient taking into account the deviation of the crystal no difficulties. shape from spherical. Since the thermodynamic Surface tension can be calculated by using the for- potentials of both phases are equal at the transition mula relating the surface-tension tensor and surface point, Eq. (1) determines the implicit dependence of energy [19] in the form the polymorphous-transformation pressure on crystal dimensions [10]. α ∫ imdUimdS The surface energy σ(hkl) was calculated in the ∂σ σ = ------S Ð S------+ µ Γ , (7) zeroth approximation, i.e., with no allowance for sur- dS ∂S j j face distortions of the interionic distances. In accordance with the definition of the surface energy [18], we where dUim is the deformation-tensor differential, S is µ Γ have at 0 K the surface area, and j and j are the chemical potential and adsorption of the jth component (over the doubly ∞ repeating subscripts i, m = 1, 2, and j = 1, 2, …, k is the σ() ()()i ()i () hkl = ∑∑ W j Ð W∞ n j hkl , (2) number of the components in the system over which the i j = 0 summation is performed). For an isotropic surface and

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 POLYMORPHOUS TRANSFORMATIONS 293

Table 1. Surface energy and surface tension for alkali halide crystals σ × 10Ð3, J/m2 α × 10Ð3, N/m Crystal B1 phase B2 phase B1 phase B2 phase 1 [20] 2 [21] 3 4 [22] 5 6 [20] 7 [21] 8 9 [22] 10 LiF 465 289 392 461 537 1491 1978 1250 2131 1997 LiCl Ð 230 271 332 318 Ð 1711 1500 1510 1481 LiBr Ð 207 244 301 309 Ð 1637 1119 1150 1107 NaF 287 266 279 356 337 521 1214 637 708 691 NaCl 280 211 260 337 318 462 641 553 621 602 NaBr 189 192 174 220 231 267 534 306 376 384 KF 233 226 257 313 320 489 719 669 720 718 KCl 177 175 185 234 228 316 404 381 444 420 KBr 170 159 162 210 211 299 341 376 428 436 RbF 224 213 237 320 308 381 600 665 712 725 RbCl 275 166 191 227 224 316 549 531 600 613 RbBr 178 150 140 215 202 286 282 271 341 350 Note: The results of the present study are listed in columns nos. 3, 5, 8, and 10. for the surface with a rotational symmetry not lower Table 2 shows that, for all the alkali halide crystals than of the third order, Eq. (7) takes the form (except for lithium fluoride), the pressure of polymor- ∂σ phous transformation increases with the reduction of σα µ Γ = Ð S------+ j j. (8) the initial crystal dimensions. For a nonspherical crys- ∂S tal (k = 2), the transition pressure is on average 6Ð8% For the one-component system, the third term in the higher than for an ideal spherical crystal (k = 1). For a right-hand side of Eq. (8) goes to zero if one uses the lithium fluoride crystal, the situation is opposite—with Gibbs equimolar interface (Γ = 0). For such a choice of a decrease in the crystal size the pressure of the B1–B2 interface, we have transition also decreases. This is explained by the different cohesion energies of the crystal, which for lith- ∂σ ium fluoride is considerably higher than for other σα= + S------. ∂S halides. This, in turn, gives rise to a higher pressure of polymorphous transformation for an infinitely large If the distance between the particles is R, then dS = LiF crystal. However, the calculation of the surface 2RdR, and, therefore, energies of ionic crystals under high pressures show R∂σ [22] that the surface energy of lithium fluoride under ασ= + ------. (9) these conditions is lower in the B2 phase than in the B1 2 ∂R RR= 0 phase and, therefore, the surface contribution to the Substituting equations of the surface energy of an thermodynamic potential accelerates the phase transi- undistorted crystal into Eq. (9), we arrive at tion by reducing the transition pressure. For the remain- ∂σ ing alkali halide crystals, the size effect results in an α()1 ()() β()i ()i R increase in the pressure of the B1–B2 phase transition hkl = ---n0 hkl ∑ Ð 1 W∞ + --- ∂------. (10) 2 2 R RR= 0 with a decrease in the crystal dimensions. i Table 1 lists the surface energy and the surface tension calculated by Eqs. (6) and (10) for a number of ELASTIC CONSTANTS OF THE B1 alkali halide crystals. Upon the determination of the MODIFICATIONS OF ALKALI HALIDE surface contribution to the thermodynamic potential of CRYSTALS the crystal, one can also calculate the pressure of the polymorphous B1–B2 transformation by the equations The occurrence of the structural phase transitions of potentials GB1 and GB2 for both crystalline modifica- makes it necessary to study the elastic properties of tions [10]. This calculation shows that the transition crystals in the high- and low-pressure ranges. The cal- pressure is strongly dependent on the crystal dimen- culations of the elasticity moduli Cαβ and their depen- sions. Table 2 lists the calculated dependences p0(r) for dence on pressure provide important information on the a number of alkali halide crystals. nature of binding forces in the high- and low-pressure

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Table 2. Pressure of polymorphous transformation (kbar) as a function of the crystal dimensions × Ð10 R0 10 , m Crystal 25 50 75 100 125 150 200 250 LiF 1 151 202 221 230 236 250 258 264 2 166 224 235 246 254 260 268 271 LiCl 1 220 198 182 176 172 171 170 169 2 231 215 202 188 182 178 176 174 LiBr 1 158 142 130 123 119 117 115 113 2 160 147 134 130 124 120 118 116 NaF 1 296 250 219 198 182 170 164 160 2 310 260 225 204 187 175 170 168 NaCl 1 174 161 144 143 142.8 142.6 142.3 142 2 183 170 151 146 145.5 145 144.7 144.5 NaBr 1 61 50 48 46 45 44.6 44.4 44 2 67 58 54 53 52.5 52 51.5 51.4 KF 1 134 120 110 108 106 104 101 100 2 140 127 116 114 113 112.7 112.5 112 KCl 1 43 39 36 34 33.5 33 32.3 32 2 51 46 41 37 36.5 36 35.7 35 KBr 1 46 40 38 37 36 35 34.5 34 2 50 44 40 38 37 36 35 34.8 RbF 1 48 44 41 39 38 37.8 37.3 37 2 51 46 43 41 40 39.5 39.2 39 RbCl 1 25 23 22 21 20.8 20.6 20.4 20.3 2 27 25 24 23.8 23.6 23.4 23.2 23 RbBr 1 20 19 18 17.5 17 16.5 16.2 16 2 22 20 19 18.5 18 17.8 17.6 17.5 Note: 1 indicates that the crystal has a spherical shape (k = 1); 2 indicates that k = 2. phases (B2 and B1, respectively) of the crystals studied When calculating the elastic constant, one has to and also allow one to draw some conclusions about the separate the contributions from the Coulomb and short- reasons for the loss of lattice stability upon compres- range forces, sion. Thus, a decrease in C44 upon crystal compression Coul sh indicates that the resistance to the shear deformation C11 = C11 + C11 . (11) decreases, i.e., the lattice loses its stability under hydrostatic compression precisely because of the loss of its The same is true for the constants C12 and C44. The resistance to shear deformation. Coulomb contribution is determined by summing up Important information on the nature of binding the interactions between all the lattice ions, whereas the forces in the B1 phase of alkali halides can be obtained short-range contribution is limited only to seven coor- by studying elastic constants C11, C12, and C44. Since dination spheres. Blackman [24] obtained the following these constants depend on the ﬁrst and second deriva- relationships for the elastic constants of cubic ionic tives of the short-range potential U(R), allowing for crystals higher coordination spheres can considerably change the calculated data. The elastic constants were calcu- 1 i 4 i 2 C = --- ∑[]Qx()+ Px(), (12) lated under zero pressure at T = 0 K. These calculations 11 V are very important, because, experimentally, the elastic i constants are usually determined by the ultrasonic method [23], which often yields dubious results, and, 1 i 2 i 2 i 2 C = --- ∑[]Qx()()y Ð Px(), (13) thus, only some experiments can be extrapolated to the 12 V region of the absolute zero. i

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Table 3. Elastic constants of the B1 modification of alkali 1 i 2 i 2 i 2 C* = --- ∑[]Qx()()y + Px(), (14) halide crystals (Pa) 44 V × 10 × 10 × 10 × 10 i Crystal C11 10 C12 10 C44 10 B 10 c2 LiF 1 9.25 5.91 5.91 8.14 C = C* Ð ---- , (15) 44 44 D 2 8.17 6.50 6.50 7.06 3 13.90 6.05 6.84 8.67 1 i i 1 c = --- ∑x y z Q, (16) LiCl 1 5.51 2.38 2.38 3.28 V i 2 5.45 2.77 2.77 3.66 3 6.07 2.27 2.69 3.54 1 i 2 i D = --- ∑[]Qx()+ Px(), (17) LiBr 1 6.12 2.20 2.20 2.51 V i 2 5.21 2.27 2.27 3.25 where V is the unit-cell volume (2R3 for the B1 phase) 3ÐÐÐÐ and (x/R, y/R, z/R) are the crystallographic coordinates. NaF 1 12.49 2.05 2.05 5.02 Summation is performed over all the lattice ions. The 2 8.08 2.78 2.78 4.55 quantities P and Q in Eqs. (12)Ð(17) are determined as 3 10.85 2.29 2.90 5.14 follows NaCl 1 5.31 1.30 1.30 2.11 1 d i () 2 4.86 1.27 1.27 2.46 P = ------UB1 r , (18) r dr R 3 6.00 1.27 1.40 2.85 NaBr 1 5.16 1.00 1.00 2.23 i () 1 d 1dUB1 r 2 4.56 1.03 1.03 2.21 Q = ------------, (19) r dr r dr R 3 4.90 0.98 1.09 2.29 where R is the equilibrium interionic distance. To cal- KF 1 7.61 1.50 1.50 3.38 culate the Coulomb components of elastic constants, 2 7.08 1.61 1.61 2.55 one has to sum up the rapidly converging series described in detail by Tosi [7]. Below, we present the 3 7.57 1.35 Ð 2.58 data calculated by the Tosi scheme: KCl 1 3.50 0.90 0.90 1.94 Coul 4 Coul 4 2 3.39 0.86 0.86 2.09 C11 ==Ð2.55604/2R , C12 0.11298/2R ,(20‡) 3 3.68 0.58 0.93 2.02 Coul 4 Coul 6 KBr 1 4.33 0.62 0.62 1.58 C44 ==1.24802/2R , D Ð4.189/2R . (20b) 2 4.16 0.70 0.70 1.86 In the calculations of the non-Coulomb components of the elastic constants, the pair potentials were approxi- 3 4.30 0.55 0.57 1.80 mated by a smooth continuous function together with RbF 1 7.11 1.21 1.21 2.80 its derivatives up to the second order. Table 3 lists the 2 6.70 1.30 1.30 3.10 elastic constants for alkali halide crystals in the B1 3ÐÐÐÐ phase. The modulus of hydrostatic compression B was calculated as RbCl 1 4.26 0.48 0.48 1.80 2 4.18 0.69 0.62 1.85 1 B = ---()C + 2C . 3 4.50 0.52 0.50 1.85 3 11 12 RbBr 1 3.90 0.47 0.47 1.45 Our calculations show that, under high pressures, 2 3.74 0.56 0.56 1.62 the properties of the B1–B2 phase transition for most alkali halides can be considered in the approximation of 3 3.97 0.40 0.41 1.59 pair-additive interactions between ions, with the multi- Note: 1 indicates the results of the present study; 2—Cohen and particle effects being ignored. The latter can be esti- Gordon results [9]; 3—experiment [23]. mated by calculating the deviation from the equality for the Cauchy relationship C11 = C44. In cubic crystals, where in the absence of external pressure the ions are account the multiparticle interactions in ionic crystals the centers of inversion, the neglect of the zero vibra- within the theory of inhomogeneous electron gas. How- tions makes the Cauchy relationship valid. Of course, ever, in many instances, the two-particle approximation our model satisfies these requirements, which is yields rather accurate results. reflected in Table 3—the relationship is satisfied for all Two difficulties can arise when comparing the theo- the crystals studied. It is rather difficult to take into retically calculated and the experimentally measured

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 296 KARPENKO et al.

C11, kbar In many instances, the elastic constants calculated at 0 (a) K are compared with the experimental data obtained at 3500 much higher temperatures. Therefore, prior to their comparison, one has to extrapolate the experimental data to the region of absolute zero, which inevitably 2500 introduces some additional errors. Figure 1 shows the elastic constants of a sodium chloride crystal as functions of the applied pressure. To 1500 construct the curves of elasticity moduli as functions of pressure Cαβ(p), we used the calculated dependences of the interionic distance and pair potential as functions of 500 the external pressure calculated in the process deter- 0 mining the pressure of the polymorphous transforma- 80 160 240 320 p, kbar tion. Minimizing the thermodynamic potential (1), we C44, kbar determine the interionic distance at the given pressure 140 r R p . The value thus obtained is then used to deter- (b) = ( ) mine the pair potential U(r) = U[R(p)], which is approximated by a smooth continuous function and its 100 derivatives up to the second order. Using the values of R and U at a given pressure, we calculate the values of P in Eq. (18) and Q in Eq. (19), which, in turn, allows us to determine the elasticity moduli under the pressure p. 60 Then, this procedure is repeated for the next pressure value p' = p + ∆p, etc. up to the pressure of the B1–B2 transformation. 20 Analyzing the curves in Fig. 1, one can see that, with 0 an increase in the pressure, the deviation from the 80 160 240 320 400 480 p, kbar Cauchy relationship increases approximately by the C12, kbar linear law γ (c) C12 Ð C44 = p, (21) 3000 where p is the applied pressure and γ = 2.85. It should be noted that Cohen and Gordon [9] obtained the value γ ≈ 2 in the approximation of three coordination 2000 spheres.

CONCLUSIONS 1000 The study of the structural phase transitions in the crystals of the composition M+X– is based on the 0 approximation of the pair ion interactions with the pair 80 160 240 320 p, kbar potential being determined by the self-consistent method within the theory of inhomogeneous electron gas. The rather good agreement between the theoreti- Fig. 1. Dependence of (a) C11, (b) C44, and (c) C12 on the cally calculated and the experimental data confirms the applied pressure for the B1 phase of sodium chloride. Dashed line indicates the calculations for the model of near- validity of the used pair potentials of ion interactions. est neighbors; solid line indicated the results of the present To confirm the accuracy of the performed calculations, study. it is very important to determine the modulus of the hydrostatic compression B and elastic constants Cαβ , because these values have already been calculated by elastic constants. First, the values of the elastic con- many researchers and are quite reliable. Thus, the good stants are somewhat ambiguous, especially that of C12. agreement between the elastic characteristics of twelve The constants C11 and C44 can be determined directly alkali halide crystals confirms the validity of the pair- by measuring the longitudinal and transverse particle wise approximation for the description of the B1 phase displacements with respect to the [100] direction, of alkali halides. The deviation from the Cauchy rela- whereas C12 can be determined only as a linear combi- tionship observed for the B2 phase under high pressure nation of elastic constants determined, as a rule, for the limits the use of two-particle description and shows [110] direction. The second difficulty is more serious. that, along with pair components of the binding forces

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 POLYMORPHOUS TRANSFORMATIONS 297 at high pressures (about one hundred kbar) in the crys- 12. A. A. Popov, V. N. Udodov, and A. I. Potekarev, Izv. tals studied, an important role is also played by nonpair Vyssh. Uchebn. Zaved., Fiz., No. 9, 80 (1999). (e.g., covalent) ionic interactions. 13. R. S. Berry and B. M. Smirnov, Zh. Éksp. Teor. Fiz. 117 (3), 562 (2000) [JETP 90, 491 (2000)]. 14. É. I. Éstrin, Materialovedenie, No. 9, 11 (1999). REFERENCES 15. S. A. Kukushkin and A. V. Osipov, Zh. Éksp. Teor. Fiz. 1. P. W. Bridgman, Proc. Am. Acad. Arts Sci. 76, 72 (1945). 113 (6), 2193 (1998) [JETP 86, 1201 (1998)]. 2. L. V. Al’tshuler, L. V. Kuleshova, M. N. Pavlovskiœ, and 16. S. V. Karpenko, A. Kh. Kyarov, and A. I. Temrokov, in G. V. Simakov, Zh. Éksp. Teor. Fiz. 47, 61 (1964) [Sov. Proceedings of the 15th International Conference Phys. JETP 20, 42 (1964)]. “Equations of State of the Matter,” Terskol, 2000, p. 42. 3. L. V. Al’tshuler, Usp. Fiz. Nauk 85 (2), 197 (1965) [Sov. 17. A. Kh. Kyarov and A. I. Temrokov, Izv. Vyssh. Uchebn. Phys. Usp. 8, 52 (1965)]. Zaved., Fiz., No. 6, 3 (1994). 18. J. W. Gibbs, The Scientiﬁc Papers, Vol. 1: Thermody- 4. R. B. Jacobs, Phys. Rev. 16 (1), 837 (1938). namics (Longmans, Green, New York, 1906; Inostran- 5. I. E. Mayer, J. Chem. Phys. 60, 270 (1974). naya Literatura, Moscow, 1953). 6. M. Born and K. Huang, Dynamical Theory of Crystal 19. S. N. Zadumkin and A. I. Temrokov, Izv. Vyssh. Uchebn. Lattices (Clarendon, Oxford, 1954; Inostrannaya Liter- Zaved., Fiz., No. 9, 40 (1968). atura, Moscow, 1958). 20. G. V. Dedkov and A. I. Temrokov, Izv. Vyssh. Uchebn. 7. M. P. Tosi, J. Phys. Chem. Solids 24, 965 (1963). Zaved., Fiz., No. 2, 19 (1979). 8. R. G. Gordon and I. S. Kim, J. Chem. Phys. 56, 3122 21. G. C. Benson and K. S. Iun, J. Chem. Phys. 42 (9), 3085 (1972). (1965). 9. A. J. Cohen and R. G. Gordon, Phys. Rev. B 12 (8), 3228 22. L. Szasz, Z. Naturforsch. A 32, 252 (1977). (1975). 23. G. Simmons and H. Wang, Single Crystal Elastic Con- 10. V. F. Ukhov, R. M. Kobeleva, G. V. Dedkov, and stants and Calculated Aggregate Properties (Cambridge A. I. Temrokov, Electronic-Statistical Theory of Metals Univ. Press, Cambridge, 1971), p. 123. and Ionic Crystals (Nauka, Moscow, 1982). 24. M. Blackman, Proc. Phys. Soc. London, Sect. B 70, 827 (1957). 11. S. N. Val’kovskiœ, V. N. Erofeev, G. I. Peresada, and E. G. Ponyatovskiœ, Fiz. Tverd. Tela (St. Petersburg) 34 (2), 360 (1992) [Sov. Phys. Solid State 34, 192 (1992)]. Translated by L. Man

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Crystallography Reports, Vol. 47, No. 2, 2002, pp. 298Ð307. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 333Ð343. Original Russian Text Copyright © 2002 by Kuznetsov, Ral’chenko, Varnin, Polyakov, Khomich, Rukovishnikov, Temirov, Teremetskaya, Tkal’, Anisimov.

REAL STRUCTURE OF CRYSTALS

Study of Polycrystalline Diamond Layers Deposited from Gas Phase G. F. Kuznetsov*, V. G. Ral’chenko**, V. P. Varnin***, V. I. Polyakov*, A. V. Khomich*, A. I. Rukovishnikov*, Yu. Sh. Temirov*, I. G. Teremetskaya***, N. V. Tkal’****, and V. G. Anisimov**** * Institute of Radio Engineering and Electronics, Russian Academy of Sciences (Fryazino Branch), pl. Vvedenskogo 1, Fryazino, Moscow oblast, 141120 Russia e-mail: [email protected] ** Center for Scientiﬁc Research, Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, 117942 Russia *** Institute of Physical Chemistry, Russian Academy of Sciences, Leninskiœ pr. 31, Moscow, 117915 Russia **** Velikiœ Novgorod State University, Velikiœ Novgorod, Russia Received, April 12, 2001

Abstract—Crystallite size in polycrystalline diamond layers with a grain size exceeding 3 µm are determined by the X-ray topography method with the use of a divergent beam from a point source. For layers with thicknesses in the range 80Ð700 µm deposited in SHF plasma and 1Ð40 µm obtained by the method of a hot ﬁlament, the size distribution of crystallites is obtained. Asterism of some spots on X-ray diffraction patterns from the diamond layers with thicknesses exceeding 100 µm showed plastic deformation of individual crystallites. The parameters of deep levels in the band gap of undoped high-resistance diamond layers and the acceptor-type defects with an activation energy higher than 1 eV are determined by the method of charge relaxation spectroscopy. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION area of Debye rings on X-ray diffraction patterns [8]. However, in polycrystalline diamond layers, the aver- Usually, the degree of perfection and the dimensions aging of crystallite size is inadmissible because crystal- of polycrystalline diamond layers deposited from the lites start growing at deliberately introduced 5 nm-large gas phase are established by transmission and scanning nuclei so that, in 0.5Ð1.0 mm-thick layers, some crys- electron microscopy (TEM and SEM) and atomic-reso- tallites attain a size of up to 1 mm. lution electron microscopy [1Ð3]. The standard method of studying the phase composition is Raman spectros- We believe that it is most convenient to study crys- copy (RS) [1, 4]. The structure of diamond layers is also tallite size in thick (ӷ1 µm) polycrystalline diamond characterized by X-ray diffractometry [5] and X-ray layers by the most informative and nondestructive X-ray topography, which provide, e.g., the study of crack for- topographic method, which, unlike SEM, does not mation in homoepitaxial diamond layers [6]. The dif- require time- and labor-consuming sample thinning. fractometric studies usually allow qualitative phase SEM provides information only about the shape and analysis which reveals, along with the phase of pure size of the faces of those crystallites which emerge to diamond, the transition layers of various carbide phases the free surface or are seen on the transverse cleavage (SiC, TiC, etc.) formed due to the interaction of carbon of the grown layer. However, such data are often insuf- with the substrate material. Both X-ray diffractome- ﬁcient for understanding and explaining the macro- try and Raman spectroscopy are also used to deter- scopic characteristics of polycrystalline diamond lay- mine internal stresses in polycrystalline diamond ers. At the same time, crystallite sizes in polycrystalline layers [5, 7]. diamond layers grown under nominally the same conditions determine the strength [9], the impurity concen- This study was undertaken to determine the condi- tration [10, 11], and the optical absorption [12] of these tions for using an X-ray method to determine the crys- layers. tallite sizes in the irradiated volume of the polycrystalline diamond layers. The methods for measuring and One of the authors (G.F.K) used the X-ray methods calculating the average crystallite size in thin layers of [8] developed for high-resolution Laue diffraction pat- various polycrystalline materials, and especially of terns suggested by Guinier, Tennevin, and Fujiwara and metals, is based on the determination of the number of showed [13Ð16] the possible use of the quantitative X-ray small spots in the discrete intensity distribution over the topographic analysis of the epitaxial layers with the use

STUDY OF POLYCRYSTALLINE DIAMOND LAYERS 299 of a divergent X-ray beam from a quasi-point source. This method was aimed at determining the sizes of blocks of mosaic and the angles of their minimum and maximum misorientation in single-crystal layers grown by the method of nonisostructural heteroepitaxy [13−16]. To study thick (40Ð1000 µm) polycrystalline diamond layers, the method of a divergent beam from a quasi-point source [17Ð19] was modiﬁed for samples in which the crystallite size exceeded 3 µm. Because of the very low absorptive power of diamond, the method of a divergent beam from a quasi-point source allows one to study polycrystalline diamond layers grown almost in the whole range of thicknesses—from several micrometers to several thousand micrometers.

EXPERIMENTAL Methods of Growth and Study of Polycrystalline Diamond Layers 20 µm Single-crystal diamond layers with a thickness in µ the range 80Ð670 m were grown in an ASTeX PDS-19 Fig. 1. Electron micrograph of a characteristic growth sur- SHF plasmochemical reactor with a 8 kW-source with face of a 110-µm-thick diamond layer grown in the SHF a frequency of 2.45 GHz [20]. The working gas was the plasma. CH4/H2/O2 mixture with ratios CH4/H2 = 2.5% and O/C = 0Ð0.7%. The substrates were polished single- Size Distribution of Crystallites crystal silicon wafers 60 mm in diameter. The films in Polycrystalline Diamond Layers were deposited under a pressure of 100 torr at substrate temperature T = 700Ð900°C. Then, the diamond layers The method of a convergent beam from a quasi- were separated from the substrates and cut into about point source provides the linear resolution along the 1-cm2-large samples. The typical structure of the sur- azimuthal direction of about 7 µm in polychromatic µ face of a grown ~100- m-thick diamond layer is shown X-ray radiation [17Ð19]. In the characteristic MoKα in Fig. 1. Thinner layers (≤40 µm) were synthesized on 1 radiation (λKα = 0.70929 Å), the resolution along the silicon by the method of a hot filament with the use of 1 the methaneÐacetone mixtures at the Institute of Physi- Bragg direction is 3 µm [13Ð16]; the resolution of the cal Chemistry of the Russian Academy of Sciences. photographic plates used is also about 3 µm. This signi- Diamond starts growing at numerous nucleation cen- fies that X-ray topography allows one to record on a ters in the form of randomly oriented crystallites finally photographic plate the discrete images of reflections forming a polycrystalline diamond layer. As a rule, from individual ≥3-µm-large crystallites. For smaller crystallites grow as columns with the transverse dimen- crystallites, the reflections merge into Debye rings. sion increasing with the layer thickness. Indeed, for finely grained samples with a thickness ranging from 1 to 40 µm, the topographs usually had To study the physical characteristics and the struc- only solid rings. However, under certain growth condi- ture of polycrystalline diamond layers thus grown, we tions (low rates of nucleation and secondary nucle- used a number of electrophysical (charge deep-layer ation), some crystallites in the layers with the thickness spectroscopy) and optical (optical absorption and d ≤ 40 µm attained dimensions sufficient for the forma- Raman spectroscopy) methods and also X-ray topogra- tion of a discrete intensity distribution not only on the phy and diffractometry. On the basis of the method of a Debye rings formed in the characteristic radiation but divergent beam from a quasi-point source [13Ð16], an also on reflections formed in the polychromatic X-ray original X-ray topographic method [17Ð19] for the radiation. quantitative analysis of crystallite sizes in the bulk of polycrystalline diamond layers irradiated with an X-ray The crystallite sizes were calculated based on their beam was developed at the Institute of Radio Engineer- sizes determined along the azimuthal direction on topographs [17Ð19]. Using a projector or an optical micro- ing and Electronics. This method was the main method ° in the present study as well. The preferable orientation scope, we measured all the reflections in the 60 -large of crystallites was studied by a more efficient X-ray dif- sector of a Debye ring formed in the diffraction of the MoKα and CoKα radiations or the polychromatic fractometry method [16, 19, 21] (Bragg reflection, 12, 12, CuKα radiation). radiation. When calculating the size of each crystallite, 1

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300 KUZNETSOV et al.

Number of crystallites zian functions and the nth-order polynomial (solid, 120 dash-dotted, and dashed lines in Fig. 2b). The maximum number of crystallites have a size not exceeding 100 µ µ (a) 31 m for the Gaussian distribution, 30.5 m for the 80 Lorentzian distribution, and 24.5 µm for the approximation by the nth-order polynomial. Thus, it is seen 60 that all three approximations give close results. It 40 should also be indicated that the histogram has two additional weak maxima (150 and 250 µm) for larger 20 crystallites. These maxima are also seen on the approximation by a polynomial. The existence of crystallites two or three times larger than the thickness of the 0 30 60 90 120 150 180 210 240 270 grown layer (80 µm) can be the consequence of the intergrowth of several neighboring crystallites with (b) close orientations. Sample 3 with a thickness of 200 µm 120 yields, in addition to the main maximum corresponding to 60 µm, also two maxima at 175 and 250 µm (Fig. 3). Some crystallites in this sample attain a size of 450 µm. 100 For a 230 µm-thick layer grown under different conditions, the main maximum corresponds to a size of about 80 20 µm (Fig. 4). Moreover, there are also two additional maxima at 90 and 160 µm that show that the position of the main maximum on the size distribution of crystal- 60 lites depends, to a large extent, on the growth conditions rather than on the layer thickness. 40 We also analyzed the layers with thicknesses of 670 and 700 µm (samples 6 and 7; sample 7 was synthe- 20 sized by the method of an arc plasmotron, Norton, USA). The size distribution of crystallites in a 700-µm- thick layer is shown in Fig. 5 (approximation by a sev- 0 enth-order polynomial). The Gaussian and Lorentzian 50 100 150 200 250 µ approximations have the main maximum at about 35 Crystallite size, m and 38 µm, respectively. Thus, in the samples studied, the maximum on the size distribution of crystallites Fig. 2. (a) Histogram of the size distribution of crystallites attains values from 5 to 40% of the layer thickness, in an 80-µm-thick diamond layer and (b) the approximation µ of this distribution by the Gaussian (dashÐdotted line) and although in layers with thicknesses of up to 200 m, Lorentzian (dashed line) functions and a seventh-order there is also a small number of crystallites two or three polynomial (solid line). times larger than the layer thickness. Since the method used in our studies does not allow the detection of crystallites with sizes less than 3 µm, they are ignored in the we took into consideration the divergence of a beam distributions obtained. from a quasi-point source and the geometry of the experiment [13Ð16]. In order to obtain the true crystallite size, it is sufficient to subtract from the crystallite Detection of Plastic Deformation length along the azimuthal direction the length of the of Crystallites image of a pointlike crystallite along the same direction µ Some diffraction spots from crystallites on X-ray in the polychromatic radiation, i.e., 7 m. In this case, topographs from thick polycrystalline diamond layers we assume that the size of each crystallite is equal to the (100Ð700 µm) show asterism [8]. Asterism manifests maximum size of its azimuthal section by the system of itself in a considerable increase in the lengths of dif- diffracting crystallographic planes, {111}. Outside the fraction spots from some crystallites along the Bragg Debye rings, some crystallites can also give rise to the direction in comparison with the lengths of the same formation of diffraction reflections from the {110} and spots along the azimuthal direction. This signifies that {331} planes. the conventional diffraction spots are transformed into The crystallite sizes determined in an 80 µm-thick characteristic diffraction “tails” caused by asterism. diamond layer are shown on a histogram in Fig. 2a. The The topograph from an 80 µm-thick polycrystalline dia- maximum number of crystallites is observed in the mond layer has reflections whose sizes in the azimuthal range 15Ð45 µm, whereas their sizes range from 3 to direction are almost equal to, or slightly larger than, 240 µm. The size distribution for crystallites in this their sizes along the Bragg direction (Fig. 6a). How- sample was approximated by the Gaussian and Lorent- ever, for polycrystalline diamond layers with a thick-

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STUDY OF POLYCRYSTALLINE DIAMOND LAYERS 301

Number of crystallites Number of crystallites 60 40 50

30 40

30 20 20

10 10

0 0 50 100 150 200 100 200 300 400 Crystallite size, µm Crystallite size, µm

Fig. 3. Size distribution of crystallites in a 200-µm-thick Fig. 4. Size distribution of crystallites in a 230-µm-thick layer approximated by a seventh-order polynomial. layer approximated by a seventh-order polynomial. ness d ≥ 100 µm, the sizes of numerous reflections in mation in individual crystallites. Our measurements the Bragg direction considerably exceed their “azi- showed that the length of the spots with tails along the muthal sizes” (Fig. 6b). The pronounced asterism of Bragg direction range from 10 to 170 µm and, in some diffraction spots indicates plastic deformation [8, 17, 18] instances, attains values of 0.5Ð1.0 mm, whereas the in some diamond crystallites. We believe that this is dimension of the point image along the Bragg direction direct proof of the fact that the intercrystallite pressure in the monochromatic MoKα radiation is about 3 µm 1 exceeded the high-temperature diamond strength. As a µ result, crystallites underwent plastic deformation dur- and, in the polychromatic radiation, 7 m. In the wave- ing layer growth. At the same time, rather symmetric length units, a point of the crystal is imaged in the dif- shapes (even circular, in accordance with the shape of fraction spot along the Bragg direction in the wave- the X-ray tube focus) of reflections indicates the non- length range δλ ≈ 7.1 × 10–4 Å of the polychromatic uniform pressure distribution in the layer during its X-ray radiation. Then the length of the tail lies within growth, and the crystallites which underwent only elastic deformation of various degrees provided the forma- Number of crystallites tion of symmetric reflections. 150 Different elastic stresses acting onto individual crystallites in the layer are explained by different contacts between neighboring crystallites (either by faces or by obtuse angles), the structure and the rigidity of the intercrystallite layers, and the crystallite sizes. The 100 topographs show that with an increase in the thickness of a deposited polycrystalline diamond layer from 100 to 670 µm, the number of plastically deformed crystallites also increases. This shows that the polycrystalline 50 diamond layer became more monolithic and, the intercrystallite layers became more rigid and provided better transfer of elastic stresses from one crystallite to another. As a result, an ever increasing number of crys- 0 tallites undergo plastic deformation and show asterism 50 100 150 200 250 on the topographs. Crystallite size, µm

Taking into account some limitations, the length of Fig. 5. Size distribution of crystallites in a 700-µm-thick the diffraction tails along the Bragg direction can be polycrystalline diamond layer approximated by a seventh- considered as an approximate measure of plastic defor- order polynomial.

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(a) mum asterism. The data on asterism can be treated mathematically in the same way as in the determination 7 6 of the crystallite sizes. The distributions of crystallites depending on the asterism they provide on diffraction patterns (the size of the spots in micrometers) for four 5 polycrystalline diamond layers are shown in Fig. 7. The 4 3 maximum number of crystallites was determined for the first highest maxima almost identical for three samples. The asterism corresponding to these maxima is about 20 µm, which exceeds by about three times the length of the point image of a perfect crystal along the Bragg direction in the polychromatic radiation and cor- δλ responds to the wavelength interval of about min = 2.1 × 10–3 Å. In addition, three weaker maxima with the 2 asterism values 60, 85, and 125 µm are also formed. 200 µm 1 These maxima are especially pronounced for sample 4, with the first maximum being shifted toward the value (b) 28 µm and the second asterism maximum being only slightly lower than the first one. The second maximum and the broad third and fourth maxima are several times 4 higher than the corresponding maxima for other samples. In comparatively thin samples (<100 µm), only 3 some crystallites were in conditions favorable for noticeable plastic deformation. The ultimate strength of a natural single-crystal dia- 2 mond at room temperature depends on the type of deformation (compression, extension, shear) and varies from 9 to 190 GPa [22Ð27]. For technical diamonds, the 1 deformation is considerably lower—from 0.23 to 0.48 GPa [27]. The theoretical calculation and the experimental measurements yield approximately the 200 µm same shear strength: 121 [22] and 132 GPa [27], respectively.

Fig. 6. X-ray topographs from crystallites in diamond layers The unit-cell parameters of silicon (a0 = 5.4282 Å) µ with the thickness (a) 80 and (b) 200 m. The crystallites and diamond (a0 = 3.5676 Å) considerably differ, and corresponding to spots 1 and 5 of the thin layer and spot 2 of the lattice mismatch for this system is about 42%. It is the thick layer have practically no asterism. Diffraction spots 2, 3, 6, 4, and 7 due to the crystallites of the thin layer well known that the growth of diamond on a single- and spots 1, 3, and 4 due to crystallites of the thick layer crystal silicon substrate is accompanied by the forma- show an increasing degree of asterism. tion of a silicon carbide interlayer, which slightly decreases the mismatch. In trying to reveal the physical reasons for the for- δλ = (1.0 × 10–3)–(17.0 × 10–3) Å and, in some instances, δλ mation of enormous elastic stresses giving rise to plas- attains a value of max = 0.1 Å. tic deformation of individual crystallites in a growing We believe that asterism is caused mainly by the polycrystalline diamond layer, one should not ignore misorientation of the individual parts of a crystallite possible isostructural heteroepitaxy of diamond crys- that underwent shear plastic deformations. The calcula- tallites on the single-crystal silicon substrate. Consider tion by the Bragg equation yields the minimum aster- one of possible models. ism of the order of one angular minute for weakly Prior to the growth of polycrystalline diamond lay- deformed crystallites. For strongly deformed crystal- ers, the surface of the single-crystal silicon substrate is lites, the misorientation attains up to 14.5′′, and, in seeded with about 5-nm-large micronuclei most proba- some rare instances, even 1.5°. Using the rough model bly randomly oriented with respect to the substrate lat- of small-angle dislocation subboundaries [22], one can tice. However, at the initial stage of growth, each of the calculate the residual elastic stresses in individual plas- micronuclei can be affected by the local mechanism of tically deformed crystallites [23]. Our calculations isostructural heteroepitaxy via the intermediate silicon showed that elastic stresses vary from 70 kPa for crys- carbide layer. Using the notions and formulas (1.5)Ð(1.9) tallites providing the minimum asterism to 20 MPa for from [28] for stresses in isostructural heteroepitaxy crystallites providing pronounced asterism and attain a with due regard for the elastic relaxation of the system value of 0.7 GPa for crystallites providing the maxi- caused by bending deformation, the considerably dif-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STUDY OF POLYCRYSTALLINE DIAMOND LAYERS 303 ferent values of the Young’s moduli for silicon and Number of crystallites polycrystalline diamond aggregates (E1 = 176.58 GPa 800 ν and E2 = 1143 GPa [26, 27]), and Poisson’s ratios ( 1 = ν 0.215 and 2 = 0.0691, respectively), we see that in isostructural heteroepitaxy at the substrate thickness d1 = 600 3 mm and the polycrystalline diamond-layer thickness µ d2 = 600 m, the expected tensile stresses in the poly- σ ≈ σ 400 crystalline diamond layer, (xx)2 (yy)2 = 274.7 GPa, σ would exceed the tensile strength t2 = 190 GPa for a natural single-crystal diamond at room temperature. For artificially grown polycrystalline diamond layers, 200 the strength limit is lower than for a natural single-crystal diamond [26, 27]. 0 There are indications of the direct effect of the sili- 0 20 40 60 80 100 120 140 160 180 conÐdiamond contact during deposition of diamond Size of spots showing asterism, µm films and layers. Thus, it is shown [23, 24] that stress relaxation occurs via the formation mainly of planar Fig. 7. Asterism distribution for crystallites in four samples defects (microtwins, stacking faults). However, as a of polycrystalline diamond layers with different thick- rule, the formation of the pure diamond phase occurs in nesses: (ᮀ) 40 µm, (᭺) 200 µm (sample 4), (᭝) 200 µm (sam- the transition layer containing amorphous carbon and ple 3), and (᭡) 670 µm. silicon carbide [1, 24]. The internal stresses arising in thick polycrystalline diamond layers can hardly be associated only with the pronounced mismatch of the compete with one another during growth. In the final diamond and the substrate lattices; nevertheless, the lat- analysis, only those crystals survive in which the direc- tice mismatch provides the formation of the stresses tion of the maximum growth rate is normal to the sub- exceeding the threshold of the plastic deformation of strate surface. The diamond layers acquire an obvious diamond. column-like structure and the crystallite sizes gradually The internal stresses determined in polycrystalline increase [1]. Therefore, the intensity ratios for the lines diamond layers [23, 24] are obviously insufficient for on the X-ray diffraction patterns from polycrystalline the plastic deformation of individual crystallites. Map- diamond layers often differ from the ratios indicated in ping microstresses in diamond layers with the prefera- the corresponding ASTM cards for the diamond pow- ble {110} crystallite orientation by the method of con- der. We obtained the diffraction patterns for Bragg dif- focal Raman spectroscopy [7], we revealed the irregu- fraction from both surfaces of the diamond layers sep- lar arrangement of the regions with a length ranging arated from the substrate—the lower finely grained sur- from several micrometers to tens of micrometers in face adjacent to the substrate and the upper coarse- which the elastic stresses attained values of up to grained (growth) surface. The relative intensities of the 9 GPa. Most often, these regions were observed in the diffraction lines obtained by the method of X-ray rock- vicinity of the crystallite boundaries [23, 26], which ing curves [17Ð19, 33] are indicated in Table 1 for two was attributed to the formation of stresses due to the appearance of incoherent crystallite boundaries in the process of formation of dislocation rows. Table 1. Intensity ratios for the lines on the diffraction patterns In plastic deformation caused by indentation of the obtained from the finely grained lower surface contacting with the substrate and freely growing coarse-grained upper surface of natural faces of a diamond single crystal at room tem- µ perature, the generation of dislocations was observed diamond layers with thickness d equal to 80 and 700 m [26, 27, 29, 30]. The cathodoluminescence topography Intensity provided recording of slip bands consisting of rows of linear dislocations [26] in two mutually perpendicular 80 700 〈 〉 hkl 110 directions. The generation of a single dislocation lower upper lower upper ASTM* or a dislocation half-loop is an elementary event of surface surface surface surface plastic deformation. However, no attempts were made to calculate the local elastic stresses within the theory 111 100 40 100 25 100 of planar dislocation pileups for diamond single crys- 220 40 75 40 100 50 tals, despite the fact that successful attempts of this 311 30 100 20 15 40 kind have already been made in conditions providing the generation of planar dislocation pileups in single 400 10 10 25 0 10 crystals of the AIIIBV semiconductors [31, 32]. 331510655025 Proceeding to textures, we should like to indicate * For comparison, the ASTM data for powder diamond samples are that randomly oriented crystals formed on the surface indicated.

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Table 2. Concentration of the nitrogen impurity in polycrys- IMPURITIES AND DEFECTS talline diamond layers Nitrogen is the main “technological impurity” in Nitrogen concentra- synthetic diamonds, and even small amounts of nitro- Sample Thickness d, µm tion, n × 1017 cmÐ3 gen (at a level of 1017 cmÐ3) considerably change the electrical, optical, thermal, and mechanical properties 1808of the grown structures, the rate of their growth, and the 2 100 9 morphology of a growing surface. In natural diamonds 3 200 20 and diamonds synthesized under high pressure, the nitrogen concentration is usually determined from the 4 200 25 light absorption coefficient at certain wavelengths in 5 230 20 the visible IR or UV ranges [9]. The application of the 6 670 26 well-known calibrations to diamond films obtained by the deposition from the gas phase is hindered by the 7 700 2 fact that the nitrogen concentration in these diamond 84060films is lower by one to two orders of magnitude and their weak characteristic bands overlap with the absorption bands due to intrinsic defects and inclusions of 80- and 700-µm-thick layers (samples 1 and 7). Similar non-diamond carbon. We used the method of calculat- to the case of powders, the intensity of the 111 lines in ing the concentration of the substituted paramagnetic the lower parts of the layers is maximal. Nevertheless, nitrogen suggested in [20, 34] (in the samples studied, for a 700-µm-thick layer, a certain texture is observed, the impurity nitrogen was in the paramagnetic state) by i.e., the preferable growth of crystallites with the (331) considering the characteristic band with the maximum planes parallel to the substrate surface. For an upper lying in the vicinity of 270 nm and providing the determination of the nitrogen concentration in unpolished sam- freely growing surface of the layer, the maximum inten- Ð4 sity is observed either for the 220 or the 311 lines, ples within an accuracy of fractions of ppm (10 at. %). depending on the growth conditions. The (110) texture The measured nitrogen concentrations in polycrystal- is characteristic of films and layers of the optical quality line diamond layers grown under different conditions × 17 Ð3 synthesized in the SHF-discharge [7]. were in the range (2–26) 10 cm (Table 2). The transmission and reflection spectra of most polycrystalline diamond layers deposited onto silicon R, % substrates had a relatively narrow band with the maxi- 22 mum at 800 cmÐ1 in the IR range attributed to the SiC layer formed at the diamondÐsilicon interface. Figure 8 shows the reflection spectra from SiC films for two 3 samples of polycrystalline diamond layers. The thick- 21 nesses of the transition SiC layers determined from these spectra are 1.2 and 3.2 nm, respectively, whereas for the remaining samples, the thickness of this layer ranged from 0.2 to 4.3 nm. 20 The electrophysical properties of polycrystalline diamond layers were studied by the method of charge- 6 based deep-layer transient relaxation spectroscopy (Q-DLTS) [35, 36]. The method is based on the mea- 19 surement of the parameters of the transient process of the charge equilibrium restoration in the sample under the action of either an electric pulse or a pulse of absorbed light. The analysis of this process provides information on the activation energy and cross section 18 of the process, and the concentrations of various structural intrinsic or impurity-induced defects with the cor- 650 700 750 800 850 900 950 responding energy levels in the band gap of the material –1 studied. The charge-based relaxation deep-level tran- Wave number, cm sient spectroscopy (Q-DTLS) differs from the well- known capacitance-based deep-level transient spectros- Fig. 8. Reflection spectra from the surfaces of the layers on copy (C-DLTS) developed by Lang [29] and allows one the substrate side for samples 3 (dashed line) and 6 (solid line). The band with the maximum in the vicinity of to study not only semiconductor but also semi-insulat- 800 cmÐ1 is attributed to the transition SiC layer with the ing (high-resistivity) materials, to which undoped poly- thicknesses 1.2 and 3.2 nm, respectively. crystalline diamond layers are related. Moreover, when

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STUDY OF POLYCRYSTALLINE DIAMOND LAYERS 305 obtaining Q-DLTS spectra, one also uses the time (rate) Q-DLTS signal ∆Q, pC window instead of the temperature one, as in [29], 35 which allows the use of isothermal spectroscopy, which (a) 2 is characterized by lower measurement errors. We 30 obtained Q-DLTS spectra with the aid of a computer- controlled ASEC-3 system [35, 36]. The signal mea- 25 ∆ sured is described as Q = Q(t1) – Q(t2), where t1 and t2 are the times measured from the beginning of discharge 20 (relaxation). In order to obtain the spectrum (the separation of the contribution that comes into the charge ∆Q 15 6 from the change in the charge state of various types of electrically active defects), the ∆Q value is measured as 10 τ a function of the time (rate) window m = (t2 – t1)/ln(t2/t1) 5 α 7 1 at the constant ratio t1/t2 = . The functional depen- 3 5 ∆ τ dence Q( m) has maxima all of which are associated 0 with the change of the charge state of the correspond- 1234567 90 ing electrically active defects—trapping centers. The (b) ∆ τ maximum of the spectral dependence Q( m) is 80 7 located at 70 τ Ð1 σΓ 2 () 60 e pn()==m pn()T exp ÐEa/kT , (1) 50 where ep(n) is the rate of hole (electron) emission from Γ × 40 the trapping centers of the given type, p(n) = 2 1/2 2 3/2 2 3 (2π/h ) k m*(), σ is the capture cross section, T is 30 pn 1 2 the temperature, Ea is the activation energy, h and k are 20 the Planck and the Boltzmann constants, respectively, 10 6 5 and m pn*() is the effective mass of a hole (electron). The 3 measured signal is related to the emission rate as 0 1234567 8 τ ∆ []()() log m, µs QQ= 0 exp Ðe pn()t1 Ð exp Ðe pn()t2 , (2)

∞ Fig. 9. Q-DLTS spectra of polycrystalline diamond layers at where Q0 = ∫ Q (t)dt. (a) T = 295 K (the region of the A peak) and (b) T = 500 K 0 (the region of the B peak). The numbers of the samples are Using Eqs. (1) and (2), we can determine the con- indicated at the curves. centration of electrically active defects, Nt = Ð1 Ð2 ∆ ∆ ln(τ T ) as a function of T–1 for each maximum or its 4 Qmax/qA, where Qmax is the maximum value of the m DLTS signal, q is the electron charge, and A is the con- slope in the Q-DLTS spectrum. tact area. The activation energy Ea and the cross section The sensitivity in the measurements of Q-DLTS σ ∆ are determined from the Arrhenius dependence spectra with the use of an ASEC-3 system was Qmin =

Table 3. Electrical parameters of polycrystalline diamond layers containing electrically active defects of types A and B Thickness Sample ρ*, Ω cm N** , cmÐ3 E , eV σ , cm2 N , cmÐ3 *** σ**** 2 d, µm tA() a(A) (A) t(B) EaB(), eV ()B , cm 1801013 >6 × 1012 0.39 2 × 10Ð22 >5 × 1013 0.49 ~10Ð23 2 100 1012 8 × 1013 0.17 2 × 10Ð22 >1014 0.58 ~10Ð20 3 200 1.4 × 1010 5 × 1012 0.24 2 × 10Ð22 >1013 0.25 ~10Ð23 5 230 1014 1011 ÐÐ>1012 0.64 ~10Ð22 6 670 5 × 107 3 × 1013 0.08 3 × 10Ð23 >6 × 1013 1.1 ~10Ð17 7 700 5 × 109 2 × 1013 0.28 5 × 10Ð21 2 × 1014 13 × 10Ð17 * Resistivity. ** Density of recharged defects upon the application of a 1 Vµm-electric ﬁeld. *** Activation energy. **** Capture cross section.

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τ –1 –2 ln m, T vided by broken carbon bonds and vacancies. The type- –4 B defects are deeper and become active only during Ä defects Ç defects sample heating. However, their capture cross section –5 6 allows one to consider them as point defects. It should also be emphasized that in the polycrystalline diamond –6 layers studied, the concentration of acceptors is higher than that of donors, NA – ND > 0, and, thus, the samples –7 have the hole conductivity. 7 The thickest layers, 6 and 7, have considerably dif- –8 2 ferent defect characteristics. The defects associated with peak B in these samples form deeper energy levels 3 –9 7 of the acceptor type with an activation energy exceed- 6 ing 1 eV, their capture cross section is higher by five to –10 six orders of magnitude than that of thinner samples, and they also have a higher concentration. Obviously, –11 5 these acceptor defects should be related to a type other than A and B. It is possible that their formation is caused –12 by the plastic deformation of crystallites revealed from 1 the analysis of X-ray topographs from thick polycrys- –13 3 talline diamond layers. 2 As was shown by Samsonenko et al. [30], diamonds –14 with considerable degree of plastic deformation possess 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3 Ω –1 a lower resistivity (10 cm for diamonds of type IIa T, K with continuous conducting dislocation networks). Fig. 10. Arrhenius dependences for the A and B defects in At temperatures lower than 230 K, the activation polycrystalline diamond layers and the corresponding A and energy of conductivity is 0.3 eV [30]. Our measure- B peaks in the Q-DLTS spectra. The activation energies Ea ments of the temperature dependence of resistivity ρ of and the capture cross sections σ of these defects are indicated in Table 3. polycrystalline diamond layers yielded an activation energy of conductivity in the range 0.2Ð0.4 eV, with the ρ value for thicker samples (6, 7) lower by several 10–16 ä, whereas the range of the variation of the time orders of magnitude than for the remaining samples window was τ = 2 × 10–6–2 × 102 s. (Table 3). It should also be indicated that the tempera- m ture dependences of conductivity provide only inte- Figure 9 shows the Q-DLTS spectra at 295 and grated information on all the electrically active defects, 500 K, and Figure 10, the Arrhenius dependences. The whereas the DLTS method allows one to single out the experimentally determined and the calculated electro- contribution to the activation energy of conductivity physical parameters of polycrystalline diamond layers that comes from various defects with the concentrations are listed in Table 3. The characteristic feature of all the differing by six to seven orders of magnitude and to samples is the pointlike electrically active defects hav- determine their capture cross sections. ing continuous energy spectra (the background in the Q-DLTS spectra). Most probably, these defects are located at the crystallite boundaries in the disordered CONCLUSION intercrystallite space. The existence of a continuous Using the X-ray topography method providing the spectrum of states is characteristic of amorphous, in determination of the crystallite sizes exceeding 3 µm, particular, diamond-like carbon materials. we analyzed quantitatively crystallites in the polycrystalline diamond layers grown from the gas phase on sil- The defect concentrations are indicated in Table 3 icon substrates and obtained their size distributions. for a certain (common for all the samples) intensity of the applied field, which allows one to estimate the rela- The X-ray topography method developed in our tive (but not the absolute) variation of the defect con- study is applicable to both polycrystalline materials centration in the samples. All the samples are character- weakly absorbing the X-ray radiation and thin layers of ized by the presence of acceptor-type defects condition- strongly absorbing metals. ally denoted here as type-A (the A peak in the Q-DLTS The X-ray topographs obtained revealed asterism of spectra) and type-B (the B peak or its slope in the diffraction spots due to many crystallites. This phenom- Q-DLTS spectra) defects. The activation energy of enon was interpreted as the first direct proof of the fact A-defects is relatively low and does not exceed 0.4 eV that the considerable part of the crystallites had under- of the valence-band ceiling. The capture cross section gone plastic deformation during the growth of poly- of these defects is low and is practically the same for all crystalline diamond layers. The plastic deformation is the samples. Most probably, these are point defects pro- assumed to be caused by the intercrystallite pressure

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 STUDY OF POLYCRYSTALLINE DIAMOND LAYERS 307 exceeding the high-temperature strength of diamond 16. G. F. Kuznetsov, Doctoral Dissertation on Physics and during the growth of thick layers. Mathematics (Moscow, 1989). The optical spectra confirmed the existence of a 17. V. G. Anisimov, L. N. Danil’chuk, G. F. Kuznetsov, transition silicon carbide layer with a thickness of sev- et al., in Proceedings of the International Workshop eral nanometers formed during the growth of polycrys- “Silicon Carbide and Related Materials” (Novgor. Gos. talline diamond layers on single-crystal silicon sub- Univ., Novgorod, 1997), p. 64. strates. It is found that the concentration of the impurity 18. V. G. Anisimov, L. N. Danil’chuk, G. F. Kuznetsov, et al., nitrogen in the samples ranges from 2 × 1017 to 2.6 × in Scientific Works of the 1st International Seminar “Topi- 1018 cmÐ3. For the first time, the charge-based relax- cal Problems of Strength” Dedicated to V. A. Likhachev and 33th Seminar “Topical Problems of Strength,” ation spectroscopy of defects in undoped high-resistiv- Novgorod, 1997, Vol. 2, Part 1, p. 186. ity polycrystalline diamond layers was used. The parameters of deep energy levels formed by defects in 19. G. F. Kuznetsov, Physics of Semiconductor Devices the band gap of diamond are determined. It is also (Narosa Publishing House, New Delhi, 1997), Vol. II, shown that the acceptor-type defects with an activation p. 1099. energy exceeding 1 eV are formed. 20. A. V. Khomich, V. G. Ralchenko, A. A. Smolin, et al., Chem. Vap. Deposition 5, 361 (1997). 21. S. A. Semiletov, G. F. Kuznetsov, A. M. Bagamadova, ACKNOWLEDGMENTS et al., Kristallografiya 23 (2), 357 (1978) [Sov. Phys. This study was supported by the Russian Founda- Crystallogr. 23, 197 (1978)]. tion for Basic Research, project no. 01-02-16046. 22. B. K. Vainshtein, V. M. Fridkin, and V. 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Crystallography Reports, Vol. 47, No. 2, 2002, pp. 308Ð312. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 344Ð348. Original Russian Text Copyright © 2002 by Kaminskii, Butashin, Aleksandrov, Bezmaternykh, Temerov, Gudim, Kravtsov, Firsov, Seo, Hömmerich, Temple, Braud. PHYSICAL PROPERTIES OF CRYSTALS

3+ Gd3Ga5O12:Nd Crystals for a Continuous-Wave Diode-Pumped Laser Operating 4 4 4 4 in F3/2 I11/2 and F3/2 I13/2 Channels A. A. Kaminskii*, A. V. Butashin*, K. S. Aleksandrov**, L. N. Bezmaternykh**, V. L. Temerov**, I. A. Gudim**, N. V. Kravtsov***, V. V. Firsov***, D. T. Seo****, U. Hömmerich****, D. Temple****, and A. Braud***** * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia e-mail: [email protected] ** Institute of Physics, Siberian Division, Russian Academy of Sciences, Krasnoyarsk, Russia *** Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia **** Research Center for Optical Physics, Hampton University, Hampton, 23688 USA ***** CIRIL, rue Claude Bloch, BP 5133, Caen Cedex, 14040 France Received October 16, 2001

3+ Abstract—Garnet crystals of the composition Gd3Ga5O12:Nd (concentration series CNd = 1–10 at. %) were grown from flux. In terms of spectroscopy, these crystals, unlike those grown from melts, form a medium with a single activator center. For the first time, continuous-wave lasing was excited by diode pumping with the use 3+ λ λ µ 4 4 of Gd3Ga5O12 : Nd crystals at the wavelengths 3 = 1.3315 and 4 = 1.3370 m of the F3/2 I13/2 channel λ λ µ 4 and also the simultaneous generation at two wavelengths, 1 = 1.0621 and 2 = 1.0600 m, of the F3/2 4 I11/2 channel. © 2002 MAIK “Nauka/Interperiodica”. INTRODUCTION Below, we report on the growth of the concentration series of Nd3+-doped gadoliniumÐgallium garnet crys- The first studies of laser generation and spectros- 3+ tals, Gd3Ga5O12:Nd (Nd-doped GGG), the study of copy of garnets were performed on flux-grown their lasing properties under diode-laser pumping and 3+ Y3Al5O12:Nd crystals [1Ð3]. The widespread use of also compare their laser and spectroscopic data these materials in active elements of flashlamp-pumped obtained with those of crystals grown by the Czochral- lasers requires the growth of large crystals 10 cm in ski method from a melt [6]. length and 1 cm in diameter. To grow such crystals, spe- 3+ cial methods of crystallization from melts were devel- The Gd3Ga5O12:Nd single crystals were grown oped such as the Czochralski method. However, the from boron–barium fluxes of the composition 56 wt % comparison of the properties of aluminum garnets (BaO + 0.62B2O3) + 44 wt % Gd3 – xNdxGa5O12, where Y Al O :Ln3+ grown from melts (T = 1930°C) and x = 0.03, 0.1, and 0.3. The saturation temperature of the 3 5 12 m ° from flux (at temperatures 1100–1200°C) showed that fluxes was Tsat = 1200Ð1230 C; the slope of its concen- ° the latter have better spectroscopic parameters corre- tration dependence was dTsat/dn = 15 C/wt %. The sponding most closely to the model of a single activator crystals were synthesized in platinum crucibles by the center [4, 5]. Furthermore, at the crystallization of group method [7] on six “prismatic” seeds with square 3+ 2 Y3Al5O12:Nd crystals from flux, the distribution coef- bases of 1 mm and heights of 10Ð15 mm. The lateral ficient of neodymium considerably exceeds the value seed faces were formed by the equilibrium (110) and (211) planes. The seeds were cut out from selected KNd = 0.18 characteristic of the crystallization from melts. In connection with the development of semicon- high-quality Gd3Ga5O12 crystals. ductor diode-laser pumping in recent years, the interest Upon the attainment of the initial overcooling by 3Ð in crystals grown from flux with a volume of up to 4°C (corresponding to the middle of the metastable 3 3+ 1 cm is renewed (including Y3Al5O12:Nd with an zone), the flux temperature was decreased at a rate of ≥ elevated activator concentration, CNd 1.5 at. %). Such dT/dt = 1Ð6°C/day (24 h), which corresponded to the crystals are used in compact laser devices providing a growth rate of the crystals at the main stage of growth, high quality of radiation (a narrow spectral line, single- up to 0.5 mm/day. The crystal holder was reversibly mode lasing, low losses, and minimum beam diver- rotated with a period of 0.5Ð1 h. The crystallization gence). temperature ranged from 80 to 100°C; the grown crys-

3+ Gd3Ga5O12:Nd CRYSTALS FOR A CONTINUOUS-WAVE DIODE-PUMPED LASER 309

I, arb. units I, arb. units ** 50 50 (a) (a) 40 40

30 30 * 20 20 * 10 * 10 * ** * 0 0 * (b) * (b) 50 40 * 40 30 30

20 20

10 10 * * * * * 0 * 0 λ 1.06 1.08 1.10 λ, µm 1.06 1.08 1.10 , µm Fig. 2. Luminescence spectra of Nd3+ ions (concentration 3+ 4 4 Fig. 1. Luminescence spectra of Nd ions (concentration ~1 at. %, F3/2 I11/2 transitions) at 15 K in the gado- 4 4 liniumÐgallium garnet crystals (a) grown from melt and ~1 at. %, F3/2 I11/2 transitions) at 300 K from gado- liniumÐgallium garnet crystals grown (a) from melt and (b) grown from flux. Asterisks indicate the lines of the prin- 3+ (b) from flux. Arrows indicate the lasing lines. cipal center of Nd ions in a Gd3Ga5O12 crystal. tals weighed 100Ð120 g. The unit-cell parameters of the A diode-laser ATS-2440 with a nominal power of 1 W crystals were measured on a Geigerflex (Rigaku Co.) thermally stabilized to 0.1°C was used as a pumping source. Absorption and luminescence spectra of Nd3+- X-ray powder diffractometer (CoKα -radiation, 1200, 1220, and 1222 reflections). doped gadoliniumÐgallium garnet crystals were recorded on an AM510-M1 (Action Research Co.) dif- 3+ Gd3Ga5O12:Nd single crystals with a volume of up fraction spectrometer by an InSbAl detector. The tem- to 2 cm3 were used to prepare the samples in the shape perature measurements of the spectra were made on a of polished planeÐparallel plates parallel to the most CSW-202 (Advanced Research Systems) double-loop developed faces of the (110) rhombododecahedron close-cycle helium refrigerator in the range 15Ð300 K. with a thickness of 2Ð4 mm and an area of up to 2 The examination of the absorption and lumines- 50 mm . This orientation of the plates reduced the cence spectra of the Nd3+-doped gadoliniumÐgallium undesirable effect of growth striae on the lasing charac- garnet crystals grown from melt and from flux obtained teristics. Similar samples were also prepared from gar- at 300 K were practically identical. The luminescence net crystals grown from melt. The ends of the active spectra consist of homogeneously broadened lines laser elements had no antireflection coating. (Fig. 1) whose positions and intensities are adequately The laser experiments were performed in a 20-cm- described within the approximation of a single activator long hemispherical cavity at 300 K. The curvature center with the scheme of crystal-field splitting sug- radius of the spherical (exit) mirror was 100 mm. The gested in [6]. At the same time, the low-temperature crystal was located close to the plane selective mirror luminescence (Figs. 2, 3) and absorption (Fig. 4) spec- through which the end pumping was made. This mirror tra were different: the number and intensities of addi- had a high transmission coefficient (í ≈ 80%) at the tional lines in comparison with those of the principal λ ≈ µ 3+ pumping wavelength ( p 0.81 m) and a high reflec- activator center Nd for crystals grown from melt were tion coefficient (R > 99.9%) at the lasing wavelengths. much more pronounced.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 310 KAMINSKII et al.

I, arb. units I, arb. units * (a) (a) 0 4I 2P 4I 4F 20 9/2 1/2 9/2 3/2 0 1 1 15 * * 2 2 10 3 3 * 4 * * 5 4 * * * 5 * 0 5

* (b) 12 (b) 0

10 1 0 * 8 * 2 1

6 3 2

4 * 4 3 * * 2 5 * 4 * * * * 0 0.86 0.88 0.90 0.92 0.94 0.430 0.432 0.87 0.88 λ, µm λ, µm

Fig. 4. Luminescence spectra of Nd3+ ions (concentration 3+ Fig. 3. Luminescence spectra of Nd ions (concentration ~1 at. %, 4I 2P and 4I 4F transitions) at 4 4 9/2 1/2 9/2 3/2 ~1 at. %, F3/2 I9/2 transitions) at 15 K in the gado- 15 K from the gadoliniumÐgallium garnet crystals liniumÐgallium garnet crystals (a) grown from melt and (a) grown from melt and (b) grown from ﬂux. Asterisks (b) grown from ﬂux. Asterisks indicate the lines of the prin- indicate the lines of the principal center of Nd3+ ions in 3+ cipal center of Nd ions in Gd3Ga5O12 crystal. Gd3Ga5O12 crystal.

These differences can be interpreted based on the All the absorption and luminescence spectral lines crystal chemistry of garnets. It is well known that in the should be attributed to the activator centers of Nd3+ ions 3+ quasi-binary Gd2O3ÐGa2O3 system, the gadoliniumÐ replacing the Gd ions in the c-positions of the struc- gallium garnet has a range of existence of ture (the concentration of large Nd3+ ions in the octahe- Gd3 + xGa5 − xO12 solid solutions enriched in gadolinium dral positions of the garnet is very low even in the oxide (Gd O ) in comparison with the stoichiometric 3+ 2 3 Nd3Ga5O12 crystals [11]). The structure of Nd activa- composition Gd3Ga5O12. The formation of these solid tor centers in the gadoliniumÐgallium garnet is deter- solutions is interpreted as a partial replacement of Gd3+ mined to a large extent by the distribution of the cations by Ga3+ ions in the octahedral positions of the structure from the second coordination sphere located in four [8, 9]; the unit-cell parameter of these solid solutions is distorted cubes, four octahedra, and six (2 + 4) tetrahe- described by the following formula [10]: dra sharing edges and vertices with a distorted NdO8 cube (see table). r()Gd3+ a = a 1 +,------1Ð × 0.0268x (1) s.s 0 3+ The principal center (type 1 in table) is formed by r()Ga distorted NdO8 cubes built by ten Ga and four Gd atoms where a0 = 12.375 Å is the unit-cell parameter of the from the second coordination sphere, which corre- stoichiometric garnet, Gd3Ga5O12, and r are the octahe- sponds to the stoichiometric garnet of the composition 3+ dral ionic radii of the ions. The existence range of solid Gd3Ga5O12 and a low concentration of Nd activator solutions is rather large (up to x ≈ 0.3). The most homo- ions. The additional activator center (type 2) observed geneous crystals are grown from a congruent melt of in the spectroscopic experiment is, most likely, formed the composition Gd3.05Ga4.95O12 [10]. in the case where nine Ga atoms and ﬁve Gd atoms are

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 3+ Gd3Ga5O12:Nd CRYSTALS FOR A CONTINUOUS-WAVE DIODE-PUMPED LASER 311

3+ within the second coordination sphere around a Nd Cation distribution around distorted NdO8 cubes within the ion. Since the octahedral ionic radius of Gd3+ (0.94 Å) second coordination sphere in Nd3+-doped gadoliniumÐgal- substantially exceeds that of Ga3+ (0.62 Å), this lium garnet replacement strongly distorts the oxygen environment Type of c-positions a-positions d-positions 3+ of a Nd ion (since the NdO8 polyhedron and the near- center (distorted cube) (octahedron) (tetrahedron) est octahedron share an edge). The calculation shows that in a gadoliniumÐgallium garnet of the congruent 1 4Gd 4Ga 6Ga composition Gd3.05Ga4.95O12 grown from melt, the cen- 2 4Gd 3Ga + 1Gd 6Ga ters of type 2 would comprise up to 8% of the total 3 3Gd + 1Nd 4Ga 6Ga number of the Nd3+ activator centers even if one ignores possible correlations characteristic of the garnet structure (the structure with large Ln atoms in c- plete absorption of the pumping energy at a wavelength positions is stabilized by the ions larger than Ga3+ of the active element of about 2Ð3 mm. located in the octahedral a-positions, e.g., by Sc, Lu, Comparing the measured spectral and laser charac- and Yb ions [12]). teristics of the Nd3+-doped gadoliniumÐgallium garnet The gadoliniumÐgallium garnet crystals grown from crystals grown from melt and from flux, we see that the ± melt studied have the unit-cell parameter aCz = 12.385 latter have somewhat narrower (by 10Ð30%) absorption 0.001 Å; their composition estimated by Eq. (1) is and luminescence lines (Figs. 2Ð4). In our opinion, the Gd3.06Ga4.94O12. Our garnet crystals grown from flux high structural disorder in crystals grown from melt is ± explained by the following: (1) The equilibrium con- have the unit-cell parameter aflux = 12.377 0.001 Å, their composition is close to stoichiometric, centration of point defects (vacancies and atoms in interstitial and unusual positions) is high at tempera- Gd3Ga5O12, which is typical of crystals grown from ° flux [13]. This explains a considerably lower concentra- tures close to 1800 C and decreases only slightly dur- tion of the second activator center (type 2) in our ing fast cooling upon completion of growth; (2) the presence in the melt of the component Ga O that can Gd Ga O :Nd3+ crystals grown from flux in compari- 2 3 3 5 12 be evaporated and dissociated with an increase in the son with its concentration in the crystals grown from temperature; and (3) the pulling rate is 1Ð2 mm/h, melt. which is higher by several orders of magnitude than the The pair centers (type 3)—a pair of Nd3+ ions in the rate of crystallization from flux. On the other hand, adjacent distorted cubes—can evidently be formed in growth from flux proceeds at a temperature of about crystals grown from both melt and flux depending on 1200°C in air, with the concentration of point defects the activator concentration in the crystal. However, the being lower and the Ga2O3 component more stable. spectra obtained show that, at the activator content of Finally, the partial filling of the octahedral positions in about 1 at. % in Nd3+-doped gadoliniumÐgarnet crys- the gadoliniumÐgallium garnet structure with Gd3+ ions tals, the concentration of these centers is lower than the also gives rise to a certain structural disorder. (Similar concentration of type 2 centers. In laser experiments, a continuous-wave lasing was 3+ Poutput, mW excited in Gd3Ga5O12:Nd crystals grown from flux. With no selective elements in the cavity, lasing 50 λ occurred simultaneously at the wavelengths 1 = λ µ 4 4 1.0621 and 2 = 1.0600 m of the F3/2 I11/2 chan- 40 3+ nel. The lasing threshold for Gd3Ga5O12:Nd crystals grown from flux was somewhat (by ~20%) lower than 30 that of crystals grown from melt. The use of selective mirrors with high reflection coefficients in the vicinity 20 of 1.33 µm highly transparent in the range 1.05Ð 1 1.06 µm in the cavity provided generation at the 2 3 λ λ µ 10 wavelengths 3 = 1.3315 and 4 = 1.3370 m of the 4 4 F3/2 I13/2 channel. The accuracy of the photoelec- ± µ 0 tric record of the lasing wavelength was 0.0005 m. 100 200 300 400 500 600 3+ The output power for the Gd3Ga5O12:Nd lasers emit- Pabs, mW ting in the vicinity of 1.06 and 1.33 µm as a function of the pumping power is shown in Fig. 5. In a crystal with Fig. 5. The emission power (at two wavelengths) of a con- ≈ 3+ µ the concentration CNd 10%, lasing in both spectral tinuous-wave Gd3Ga5O12:Nd laser in the 1.06 m range with activator concentrations (1) 3 and (2) 10 at. % and in ranges was also excited, which shows that one can use µ 3+ the 1.33 m range with activator concentration (3) 10 at. % as highly concentrated Gd3Ga5O12:Nd crystals to design a function of the pumping power of a semiconductor laser λ µ minilasers. These high concentrations provide the com- ( pump = 0.81 m).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 312 KAMINSKII et al. phenomena were earlier observed when studying the Systems.” The study was supported by the Russian crystals grown from melt in the mixed Foundation for Basic Research, International Associa- 3+ 3+ Nd :Y3(Ga,Al)5O12 [14] and Nd :Y3(Y,Ga)2Ga3O12 tion for the Promotion of Cooperation of the Former [15] systems.) However, the crystallization from melt Soviet Union (INTAS), project no. 99-01366; and the does not allow the attainment of a high degree of gado- Programs “Fundamental Metrology,” “Fundamental liniumÐgallium garnet matrix disorder because of the Spectroscopy,” and “Laser Physics” of the Ministry of substantial divergence of the solidus and liquidus Industry, Science, and Technology of the Russian Fed- curves in the instances where the composition of the eration. melt differs from the congruent one. Therefore, the crystallization from flux seems to be more advanta- geous because the range of existence of the REFERENCES Gd3 + xGa5 − xO12 solid solutions is also quite large at a 1. J. E. Geusic, H. M. Marcos, and L. G. Van Uitert, Appl. temperature of 1200°C [10]. Thus, varying the Phys. Lett. 4, 182 (1964). Gd2O3/Ga2O3 ratio in the flux, it is possible to grow a dis- 2. A. A. Kaminskii, Zh. Éksp. Teor. Fiz. 51, 49 (1966) [Sov. Phys. JETP 24, 33 (1966)]. ordered garnet of the composition up to Gd3.3Ga4.7O12. Thus, the concentration series of the 3. G. A. Bogomolova, A. A. Kaminskii, and V. A. Timofe- 3+ eva, Phys. Status Solidi 16, K165 (1966). Gd3Ga5O12:Nd garnet crystals were grown from flux. It was shown that unlike crystals grown from melt, 4. V. V. Osiko, Yu. K. Voronko, and A. A. Sobol, Crystals spectroscopically, these crystals can be considered as a (Springer-Verlag, Berlin, 1984), Vol. 10, p. 87. medium with a single activation center. For the first 5. V. Lupei, Opt. Mater. 16, 137 (2001). time, continuous-wave lasing was excited in diode- 6. Kh. S. Bagdasarov, G. A. Bogomolova, M. M. Gritsenko, 3+ et al., Dokl. Akad. Nauk SSSR 216 (5), 1018 (1974) pumped Gd3Ga5O12:Nd crystals at wavelengths λ λ µ 4 4 [Sov. Phys. Dokl. 19, 353 (1974)]. 3 = 1.3315 and 4 = 1.3370 m of the F3/2 I13/2 channel and the simultaneous generation at two wave- 7. L. N. Bezmaternykh, I. A. Gudim, and V. L. Temerov, in λ λ µ Proceedings of the 9th National Conference on Crystal lengths 1 = 1.0621 and 2 = 1.0600 m of the channel Growth, Moscow, 2000, p. 171. 4 4 F3/2 I11/2. The power characteristics of lasing can 8. C. D. Brandle and R. L. Barns, J. Cryst. Growth 26, 169 be essentially improved by using an antireflection coat- (1974). ing with optimum dimensions and the activator concen- 9. J. R. Carruthers, M. Kokta, R. L. Barns, and M. Grasso, tration or by optimizing the cavity. In the growth exper- J. Cryst. Growth 19, 204 (1973). iments, an interesting problem, along with the reduc- 10. M. Allibert, C. Chatillon, J. Mareschal, and F. Lissalde, tion in the optical inhomogeneity (growth striae and J. Cryst. Growth 23, 289 (1974). 3+ faceting) of the Gd3Ga5O12:Nd crystals, is also the 11. B. V. Mill and A. V. Butashin, Izv. Akad. Nauk SSSR, 3+ ≥ preparation of the Gd3 + xGa5 – xO12:Nd crystals (x 0.2) Neorg. Mater. 18 (8), 1394 (1982). with considerable structural disorder. The media whose 12. B. V. Mill, Magnetic and Crystallochemical Study of absorption spectra have broadened lines provide better Ferrites (Mosk. Gos. Univ., Moscow, 1971), p. 64. matching with the diode pumping whose wavelength 13. W. Tolksdorf, G. Bartles, G. P. Espinosa, et al., J. Cryst. can experience temperature drift. Growth 17, 322 (1972). 14. R. K. Watts and W. C. Holton, J. Appl. Phys. 45 (2), 873 ACKNOWLEDGMENTS (1974). The authors are grateful to B.V. Mill for measuring 15. A. A. Kaminskii, K. Ueda, D. Temple, et al., Opt. Rev. 7 (2), 101 (2000). the unit-cell parameters of the crystals. The authors greatly benefited from the cooperation with the Joint Open Laboratory “Laser Crystals and Precise Laser Translated by A. Zolot’ko

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 313Ð317. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 349Ð353. Original Russian Text Copyright © 2002 by Burkov, Denisov, Perekalina. PHYSICAL PROPERTIES OF CRYSTALS

Circular Dichroism of Uniaxial Sulfate Crystals in the Range of Electronic Transitions of Sulfate Groups V. I. Burkov*, Yu. V. Denisov*, and Z. B. Perekalina** * Moscow Institute of Physics and Technology, Institutskiœ per. 9, Dolgoprudnyœ, Moscow oblast, 141700 Russia ** Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia e-mail: [email protected] Received April 16, 2001; in ﬁnal form, November 8, 2001

Abstract—Absorption and circular-dichroism spectra of uniaxial inactivated sulfate crystals KLiSO and µ 4 [C2H4(NH2) á H2SO4] have low-intensity bands in the spectral range of 0.20Ð0.35 m. These bands are attributed to forbidden transitions in a sulfate group. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION sented. However, there is no direct experimental evi- The study of cubic double sulfates-langbeinites dence of the contribution of these transitions to the gyrotropic properties of these crystals. crystals (sp. gr. 2 3) shows that an SO 2Ð -group is one of 1 4 Below, we describe the detailed study of the absorp- the chromophores contributing to a crystal gyrotropy tion and the circular-dichroism spectra of inactivated (see review [1]). In the ultraviolet range of 0.18Ð0.30 µm, uniaxial potassiumÐlithium sulfate (KLiSO4) 1 and eth- the forbidden electronic transitions in a sulfate group in ylenediamine sulfate [C H (NH ) á H SO ] crystals 2 these crystals manifest themselves in the absorption 2 4 2 2 2 4 and circular-dichroism spectra. The main contribution and the experimental data on the involvement of elec- to the optical rotation in the range of 0.2Ð0.8 µm comes tronic transitions in sulfate groups in the gyrotropic from the allowed transitions of a more short-wave properties of these crystals. length range. The study of the optical-rotation dispersion in these µ In this connection, the role of a sulfate group in the crystals in the range 0.2Ð0.6 m shows that the disper- gyrotropy of inactivated uniaxial sulfate crystals is of sion curves are of a conventional nature [4Ð6], which great interest. The data on the gyrotropy of uniaxial shows that the contribution of forbidden transitions in sulfate groups to the optical-rotation dispersion is KLiSO4, KNaSO4, [C2H4(NH2)2 á H2SO4] (ethylenediamine sulfate) and NiSO á 6H O crystals, and biaxial insigniﬁcant. Hence, one can hardly expect the forma- 4 2 tion of intense bands in the range of 0.20Ð0.35 µm in (NH CH COOH) á H SO (triglycine sulfate) and 2 2 3 2 4 the circular-dichroism spectra. MgSO4 crystals are reviewed in [2]. Gyrotropy in these compounds manifests itself only in the crystalline state.

The effect of electronic transitions in sulfate groups STRUCTURE OF KLiSO4 on the gyrotropic properties of these crystals is of spe- AND [C2H4(NH2) á H2SO4] CRYSTALS cial interest because, unlike cubic langbeinites, the con- 6 tribution to the circular dichroism and optical rotation The structure of a KLiSO4 crystal 1 (sp. gr. P63(C6 ), in uniaxial and biaxial crystals comes not only from the unit-cell parameters a = 5.15 Å and c = 8.63 Å, and 〈pm〉 but also from the 〈pq〉 modes (where 〈p〉, 〈m〉, and Z = 2) is a close packing of tetrahedral sulfate ions with 〈q〉 are the matrix elements of the electric, magnetic, local symmetry C3 (Fig. 1) [7]. Sulfur atoms are located and quadrupole moments of the transitions, respec- in a tetrahedron with one face lying in the plane normal tively [3]). to the 63-axis (the optical axis of the crystal). Potassium An active role of electronic transitions in sulfate atoms occupy the octahedral voids between the tetra- groups in the gyrotropy of uniaxial crystalline sulfates hedra of sulfate groups, whereas lithium atoms are in is indirectly shown in [4], where the data on circular- tetrahedral voids. Potassium atoms are located on the dichroism spectra of crystalline sulfates doped with tet- 63-axis and lithium atoms, on the 3-axis. Two mole- 2Ð 2Ð cules in the unit cell related by the 6 -axis form no heli- rahedral CrO4 and åÓO4 -ions incorporated into the 3 2Ð positions of sulfate groups in the crystal lattice are pre- cal structure of SO4 -tetrahedra.

314 BURKOV et al.

a2=8.63 structure are distorted, the SÐO bond lengths in sulfate groups differ by 0.01 Å from the conventional length of 1.5 Å. It follows from the description of the structure that a1=5.15 the chromophors responsible for the optical activity in 2Ð + + a crystal 1 are SO4 -groups and also K - and Li -ions in the positions with C3 symmetry. In crystal 2, in addition to sulfate groups (C2 symmetry), a chromophor responsible for the optical activity is also ethylenediamine (C2 symmetry). Thus, it is seen that both the symmetry and the orientation of sulfate groups relative to the optical axis in crystal 2 differ from those in crys- KLiSO tal 1. In other chromophors, electronic transitions lie in 4 the range of vacuum ultraviolet and, therefore, their bands cannot be measured. Fig. 1. Fragment of the KLiSO4 structure. EXPERIMENTAL The structure of a [C H (NH ) á H Sé ] crystal 2 2 4 2 2 2 4 The dispersion of optical rotation was studied on a 4 8 (sp. gr P4122 (D4 ) or ê4322 (D4 ), unit-cell parameters spectropolarimeter. The absorption spectra were a = 8.47 Å and c = 18.03 Å, and Z = 8) (Fig. 2) consists obtained on a Specord M-40 spectrophotometer and on 2Ð a spectrophotometric setup with an MDR-23 mono- of layers of ethylenediamine and SO4 -groups normal chromator. The spectra of circular dichroism were stud- to screw axis 41, which is the optical axis of the crystal ied on a Mark-3S dichrometer. These devices were modiﬁed with the aid of the program of computeriza- [8]. The ethylenediamine (the C2 symmetry) is located in the structure in such a way that the CÐC bond is tion of nonstandard optical techniques written in the almost parallel to screw axis 4 of the crystal. The dis- Moscow Institute of Physics and Technology on the 1 basis of Pentium computers with unique interface cards torted tetrahedra of sulfate groups (the C2 symmetry) for control and data collection. This computer program are also located around the 41-axis so that the twofold not only allows one to control the operations performed axes of the tetrahedra coincide with the twofold axes of by the device and obtain experimental data and their the crystal lattice. Because the tetrahedra in the crystal graphical mapping in real time, but also provides a

y CS O N

x –1/2–1/4

0 1/2

6 .2 3 2.82 2.81 3.32 .32 3 2 .8 2.80 0 3 .2 1/2 6 3.31 –1/2

1 1 z 8 3 . .

2 3 y

Fig. 2. Two projections of the fragment of the ethylenediamine sulfate structure.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 CIRCULAR DICHROISM OF UNIAXIAL SULFATE CRYSTALS 315 complete set of procedures for spectra processing, K, cm–1 ∆ε, 10–4 cm–1 including modern methods for reducing the noise com- 6 ponents of the spectra, transformation of the spectra, and their approximation by given contours, elimination of undesirable background components, comparison of 10 spectra, etc. Using this equipment and the program of the data processing, we managed to reveal new details 4 of the spectra, in particular, in the range with pro- 2 nounced absorption. Ethylenediamine sulfate crystals were grown from 1 5 aqueous solutions by the method of spontaneous crys- 2 tallization and were of a high optical quality. The measurements were made on plates cut out parallel to the cleavage plane and oriented normally to the 41-axis coinciding with the optical axis of the crystal. The KLiSO crystals were grown from aqueous solutions on 0 0 4 0.20 0.25 0.30 0.35 a seed by the method of decreasing temperature. The λ, µm study of specimens in a polarization microscope showed the absence of any domain structure. The spec- Fig. 3. (1) Absorption and (2) circular-dichroism spectra of tra were obtained from the specimens from 0.5 to a KLiSO4 crystal. 4.0 mm in thickness. K, cm–1 ∆ε, 10–4 cm–1 RESULTS AND DISCUSSION 2.0 0.8 The absorption spectra in the range 0.2Ð0.3 µm are similar for both crystals (Figs. 3 and 4, curve 1) and resemble the corresponding spectra of double sulfate crystals (langbeinites) [1]. The results obtained show 1.5 0.6 that absorption in this spectral range is caused by elec- 2 2Ð tronic transitions in SO4 groups. The intensity of the band with the maximum at 0.28 µm is very low, espe- 1.0 0.4 cially for crystal 2, which is typical of the transitions ×0.15 forbidden by the selection rules. At the same time, the positions and intensities of the 2 bands on the circular-dichroism spectra are different. 0.5 0.2 The circular-dichroism spectra of crystal 1 (Fig. 3, curve 2) show only one low-intensity band with the maximum at λ = 0.28 µm in the range of 0.25Ð0.30 µm, 1 whereas the spectrum of crystal 2 has two bands at λ 0 0 equal to 0.30 and 0.25 µm in this range (Fig. 4, curve 2). 0.20 0.25 0.30 0.35 Moreover, the circular-dichroism spectrum of crystal 2 λ, µm has the third band with the maximum at λ = 0.21 µm Fig. 4. (1) Absorption and (2) circular-dichroism spectra of with the intensity higher by an order of magnitude than ethylenediaminesulfate crystal [curve (2) corresponds to the the intensity of the two first maxima (Fig. 4, curve 2). data reduced by a factor of 15 along the ordinate axis]. Evidently, different gyrotropic properties of these crystals in the range of electronic transitions in sulfate groups should be explained by different orientations of tional methods of quantum chemistry [12Ð17]. At these groups with respect to the optical axis along present, the following sequence of the filled orbitals is 2 6 2 6 4 which the axial circular-dichroism spectra are mea- universally accepted: (1a1) , (1t2) , (2a1) , (2t2) , (1e) , 6 6 sured. (3t2) , (1t1) [17] (Fig. 5). The antibinding orbital with Now, consider qualitatively the effect of the orienta- the minimum energy has not been definitively estab- tion of SO 2Ð -ions depending on the electronic structure lished as yet. In a number of studies, the first empty 4 * of sulfate groups. The electronic structure of tetrahedral orbital was calculated to be (4t2 ) in the sequence sulfate groups was intensely studied in the 1960Ð1970s. (4t* )(3a*)(2e*)(5t* ); in other studies, it was calculated The electronic structure of the sulfate group was first 2 2 studied in [9Ð11]. Later, calculations of the energy of as (3a1* ) in the sequence (3a1* )(4t2* )(2e*)(5t2* ). In the molecular orbitals were made by various computa- first case, because of the interaction with a hole on the

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 316 BURKOV et al.

E these groups is parallel to the optical axis of the crystal, only the AÐE transitions are active in the axial circular- dichroism spectrum.

2Ð In crystal 2, the symmetry of SO4 groups is C2, * 3a1 and, therefore, all the T states should be split into the A- and 2B-states. However, the bond lengths and the bond angles show that sulfate groups in crystal 2 have a * 4t2 higher (D2) symmetry. It is expedient to consider this group, because the directions of the moments of the 〈µ〉 〈 〉 〈µ〉 0i and m i0 transition (where 0i is the matrix ele- 3p 〈 〉 ment of the electrical dipole moment, m i0 is the matrix element of the magnetic dipole moment, and A, B, and B' are the excited i-states) within this group are aligned along three mutually perpendicular twofold axes. Thus, knowing the orientation of sulfate groups in crystal 2, 1t1 π we can state that the projections of the moments of all (e, t1, t2) 3t2 the transitions onto the plane normal to the optical axis 1e 2p have nonzero values. Therefore, the transitions from the ground A-state to all the excited A, B, and B' states may 2t δ(a , t ) 2 1 2 contribute to the axial circular-dichroism spectrum. 3s 2a1 The above consideration explains different circular- dichroism spectra for crystals 1 and 2. Thus, the studies of the circular-dichroism spectra of uniaxial inactivated crystals of double sulfates 2Ð showed that SO4 -groups are the chromophors responsible for the optical activity. This signiﬁes that both 2Ð allowed and forbidden transitions in the SO4 groups 2s(a, t ) 2 contribute to the optical activity of these crystals. In the spectral range under study (0.2Ð0.35 µm), only the forbidden transitions are active in the formation of circu- 1t2 lar-dichroism spectra dependent on the orientation of SO4 groups, whereas the allowed transitions are 1a1 responsible mainly for the dispersion of the rotation of the polarization plane at values exceeding 0.2 µm [2].

Fig. 5. Schematic of molecular orbitals of sulfate groups ACKNOWLEDGMENTS (open circles denote electrons on molecular orbitals). The authors are grateful to A.V. Egorysheva for study of the absorption spectra. 5 ligand orbital, the (1t1) (4t2* ) state yields the following 3 3 1 1 1 1 terms: T1, T2, T1, T2, E, A1. In the second case, REFERENCES because of the above interactions, the excited conﬁgu- 5 * 1 1 3 1. V. I. Burkov and Z. B. Perekalina, Neorg. Mater. 37 (3), ration (1t1) (3a1 ) yields two terms, T2 and T2. 1 (2001). 1 1 Within the Td symmetry, the A1– T2 transitions are 2. V. A. Kizel’ and V. I. Burkov, Gypotropy of Crystals allowed in the electrical dipole approximation. Because (Nauka, Moscow, 1980). of the spinÐorbital interactions, the appearance in the 3 3 1 1 3. F. S. Richardson and G. Hilmes, Mol. Phys. 30, 237 absorption spectrum of the transitions to T1, T2, T1, E, (1975). 1 A1 forbidden in the electrical dipole approximation cannot be excluded. 4. V. I. Burkov, Neorg. Mater. 30 (1), 12 (1994). 2Ð 5. Z. B. Perekalina, A. Yu. Klimova, and L. M. Belyaev, It is clear that, within the Td symmetry, SO4 -groups Kristallograﬁya 23 (1), 124 (1978) [Sov. Phys. Crystal- do not possess any optical activity. In crystal 1, the sym- logr. 23, 66 (1978)]. metry of sulfate groups is C3, and all the T-states are 6. V. A. Kizel’, Yu. I. Krasilov, and V. N. Shamraev, Opt. split into the A- and E-states. Since the threefold axis of Spektrosk. 17, 863 (1964).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 CIRCULAR DICHROISM OF UNIAXIAL SULFATE CRYSTALS 317

7. M. Karppinen, J. O. Lundgren, and R. Liminga, Acta 14. R. Manne, J. Chem. Phys. 46, 4645 (1967). Crystallogr., Sect. C: Cryst. Struct. Commun. C39, 34 (1983). 15. M. Ketton and D. P. Santry, Chem. Phys. Lett. 7, 105 (1970). 8. K. Sakurai, J. Phys. Soc. Jpn. 16, 1205 (1961). 9. D. M. Bishop, M. Randict, and J. R. Morton, J. Chem. 16. U. Gelius, B. Roos, and P. Siegbahn, Theor. Chim. Acta Phys. 45, 1880 (1966). 23, 59 (1971). 10. D. M. Bishop, Theor. Chim. Acta 8, 285 (1967). 17. G. Hojer, S. Meza-Hojer, and G. Hernández De Pedrero, 11. D. M. Bishop, J. Chem. Phys. 48, 1872 (1968). Chem. Phys. Lett. 37, 301 (1976). 12. H. Johansen, Theor. Chim. Acta 32, 273 (1974). 13. K. Taniguchi and B. L. Henke, J. Chem. Phys. 64, 3021 (1976). Translated by T. Dmitrieva

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 318Ð319. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 354Ð356. Original Russian Text Copyright © 2002 by Perekalina, Kaldybaev, Konstantinova, Orekhova, Tynaev. PHYSICAL PROPERTIES OF CRYSTALS

Influence of X-ray Irradiation on the Optical Properties of Chromium-Doped KLiSO4 Crystals Z. B. Perekalina*, K. A. Kaldybaev**, A. F. Konstantinova*, V. P. Orekhova*, and A. D. Tynaev** * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow; e-mail: [email protected] ** Issyk-Kul State University, Karakol, Kyrgizstan Received June 13, 2001

Abstract—The absorption and circular-dichroism spectra of chromium-activated KLiSO4 crystals both nonirradiated and irradiated with an X-ray beam have been studied. It was established that in nonirradiated crystals chromium ions are mainly trivalent (Cr3+) and have octahedral coordination. In irradiated crystals, along with the centers provided by (Cr3+) ions, new centers are formed associated with (Cr4+) and (Cr5+) ions. © 2002 MAIK “Nauka/Interperiodica”.

Doping crystals gives rise to the formation of cen- The absorption spectra of undoped and chromium- ters that have different local environments that manifest activated KLiSO4 crystals were studied on 1.85- and themselves in the absorption and circular-dichroism 0.95-mm-thick plates oriented normally to the optical spectra. axis in the spectral range from 200 to 800 nm at room It is well known that the irradiation of crystals with temperature (Fig. 1). an X-ray beam is usually accompanied by a change in Undoped KLiSO4 crystals are transparent in the vis- ion valence and, as a consequence, also in the physical ible and ultraviolet spectral ranges (Fig. 1, curve 1). In and chemical properties of the crystals. Circular- the absorption spectra of Cr-doped KLiSO4 crystals, dichroism spectra provided by different optically active three wide bands with the maxima at λ = 280, 420, and centers associated with the incorporation of impurities 600 nm are observed (Fig. 1, curve 2), which are char- into a crystal are extremely sensitive to changes in their acteristic of octahedrally coordinated Cr3+ ions and local symmetry caused by various external factors, associated with the allowed spin-quartet transitions including irradiation. In this connection, it was expedi- 4 4 2 ent to study the effect of the X-ray irradiation of crystals from the A2g ground state to the T2g(F), T1g(F), and 4T (P) levels of the Cr3+ ion. The absorption spectra of on the circular dichroism of chromium-activated KLiSO4 1g crystals with well-known optical properties [1, 2]. irradiated chromium-doped KLiSO4 crystals are also A potassiumÐlithium sulfate KLiSO crystal shown in Fig. 1 (curve 3). It is seen that upon irradiation 4 of these crystals, the intensity of the absorption band at belongs to the hexagonal pyramidal symmetry class 6, λ ~ 600 nm decreases, which shows that the number of sp. gr. P6 with two molecules in the unit cell. The 3 Cr3+ ions diminish. The decrease in the intensity of this KLiSO crystals have the tetrahedral framework struc- 4 band in the spectrum of the irradiated crystal is accom- ture consisting of chains of distorted oxygen tetrahedra panied by the formation of a new absorption band at with sulfur and lithium ions located on the threefold 5+ + 355 nm attributed to Cr ions. Comparing the EPR and axes and K ions located the sixfold axis in the octahe- the optical-absorption spectra, we established that dral voids formed by these tetrahedra [3, 4]. under the effect of X-ray irradiation, some Cr3+ ions in Chromium-doped KLiSO4 crystals (KLiSO4 : Cr) alkali metal crystals change their valence by gradually were grown from an aqueous KLiSO4 solution with trapping two holes according to the scheme [7] 1.5 g/l of Cr2(SO4)3 á 6H2O at the Issyk-Kul State Uni- 3+ 4+ 5+ versity [1]. Well-faceted green transparent crystals Cr + h Cr + h Cr . were used to prepare the specimens (the crystal seg- The circular-dichroism spectra of nonirradiated and ments having no stresses or domains were used) [5]. irradiated chromium-doped KLiSO4 crystals were stud- The analysis of the EPR spectra of the specimens [6, 7] ied in the spectral range 230Ð800 nm with a sensitivity showed that chromium ions in KLiSO4 crystals are of ~4 × 10–6 optical density per millimeter on a trivalent and replace K+ ions in strongly distorted octa- dichrometer (Fig. 2). hedral voids. The changed valence is compensated by The circular-dichroism spectra of nonirradiated 3Ð an SO3 -radical. specimens (Fig. 2, curve 2) show two wide bands with

INFLUENCE OF X-RAY IRRADIATION ON THE OPTICAL PROPERTIES 319

K, cm–1 ∆ε, 10–3, cm–1 2.0 3 6

1.6 4 2 2 3 1.2 0 0.8 –2

2 –4 0.4 1 –6 0 200 300 400 500 600 700 800 λ, nm 200 300 400 500 600 700 800 λ, nm

Fig. 1. Absorption spectra of undoped and chromium-activated potassiumÐlithium sulfate crystals; (1) KLiSO4; Fig. 2. Circular-dichroism spectra of potassiumÐlithium sul- (2) nonirradiated KLiSO4 : Cr; (3) KLiSO4 : Cr crystals fate crystals; (2) nonirradiated KLiSO4 : Cr; (3) KLiSO4 : Cr irradiated with an X-ray beam. irradiated with an X-ray beam. two maxima at 420 and 650 nm corresponding to the Thus, it is shown that the X-ray irradiation of 3+ Cr bands observed in the absorption spectrum. In the KLiSO4 : Cr crystals changes the valence and the local spectra of circular dichroism, these bands have oppo- environment of chromium ions. The examination of the site signs. The wide band with the maximum at 650 nm circular-dichroism spectra of crystals shows that these in the long wavelength edge has a narrow band at 685 nm spectra are more informative than the absorption spec- 4 2 which, most probably, corresponds to the A2 E tra. The study of the circular-dichroism spectra of irra- transition [2]. diated crystals provides more detailed information on the transition of the impurity from one valent state to In the circular-dichroism spectra, the additional another. bands with the maxima at 500 and 550 nm are also observed to be not associated with the Cr3+ ions. Pre- sumably, these bands are caused by tetrahedrally coor- REFERENCES dinated Cr4+ ions [8]. The absence of these bands in the 1. A. A. Alybakov, N. A. Gubanova, and K. Shaersheev, absorption spectrum of irradiated crystals seems to be Cryst. Res. Technol. 43 (2), K92 (1982). explained by the low concentration of these ions in the crystal. In the ultraviolet range of the circular-dichro- 2. B. N. Grechushnikov, T. F. Veremeœchik, K. A. Kaldy- ism spectrum (250Ð370 nm), a wide band, evidently vaev, et al., Zh. Prikl. Spektrosk. 54 (4), 669 (1991). combining the bands associated with chromium ions of 3. R. Wyckoff, Crystal Structures (Interscience, New York, 3+ + different valences (Cr , Cr ), SO4 groups, and color 1960), Vol. 3, p. 112. centers [2, 7, 9], is observed. 4. M. Karppinen, J.-O. Lundgren, and R. Liminga, Acta As is seen from Fig. 2, the circular-dichroism spec- Crystallogr., Sect. C: Cryst. Struct. Commun. C39, 34 (1983). tra of the irradiated KLiSO4 : Cr crystal (Fig. 2, curve 3) have considerably changed because of the changes in 5. H. Klapper, Th. Hahn, and S. J. Chung, Acta Crystal- the Cr3+ and Cr4+ ion concentrations and the appearance logr., Sect. B: Struct. Sci. B43, 147 (1987). of Cr5+ ions. The intensities of the circular-dichroism 6. L. Conte and F. Holuj, Phase Transit. 6, 259 (1986). bands of Cr3+ ions at λ ~ 420 and 650 nm decreased and 5+ 7. A. A. Alybakov and K. Sharsheev, Izv. Akad. Nauk Kirg. a wide band with the maximum at 355 nm due to Cr SSR, No. 3, 20 (1984). ions appeared. Moreover, the intensity of the band due to Cr4+ ions in the green spectral range considerably 8. P. I. Macfarlane, T. P. J. Han, B. Henderson, and increased and the band maximum was shifted to λ Ϸ A. A. Kaminskii, Opt. Mater., No. 3, 15 (1994). 570 nm. The positional changes observed for the band 9. V. I. Burkov, Yu. V. Denisova, and Z. B. Perekalina, Kri- maxima at 650 nm in the circular-dichroism spectra can stallograﬁya 47 (2), 313 (2002). be caused by the appearance of ions having different valence and a change of the local environment of the absorption center because of irradiation. Translated by T. Dmitrieva

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 320Ð323. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 357Ð360. Original Russian Text Copyright © 2002 by Beda, Bush, Volkov, Meshcheryakov. PHYSICAL PROPERTIES OF CRYSTALS

3+ EPR Spectra of Cr -Doped LuNbO4 Crystals A. G. Beda*, A. A. Bush**, A. F. Volkov**, and V. F. Meshcheryakov** * Institute of Theoretical and Experimental Physics, ul. Bol’shaya Cheremushkinskaya 25, Moscow, 117259 Russia e-mail: [email protected] ** Moscow State Institute of Radioengineering, Electronics, and Automation (Technical University), pr. Vernadskogo 78, Moscow, 117454 Russia Received April 2, 2001

3+ Abstract—Ferroelastic LuNbO4 single crystals containing 0.3 at. % Cr ions have been grown by the ﬂoating zone technique, and their EPR spectra have been studied at a frequency of 9.8 GHz at room temperature. The lines on the spectra are due to the transitions caused by three paramagnetic centers formed as a result of the replacement of one isovalent position of a Lu3+ ion and two nonisovalent positions of Nb5+-ions by Cr3+-ions. As a result of twinning, the line number is doubled and four principal directions arise along which the same spectra are obtained. The spectra of these centers were described by a spin Hamiltonian with S = 3/2, the D Ð1 and E parameters ranging from 0.024 to 0.17 cm , and the g-factors g|| = 1.75–2.20 and g⊥ = 1.90–2.13. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION structure in which tungsten atoms are surrounded with The use of the method of dynamic orientation of the four oxygen atoms forming a regular tetrahedron, in the nuclei suggested in [1] requires the selection of a suit- fergusonite structure, two additional oxygen anions are able working substance with a pronounced quadrupole located close to a niobium atom. As a result, the coor- splitting of the energy levels of the nucleus studied (the dination number of niobium is (4 + 2). The LuO8-poly- NQR frequency should range within several dozens of hedra share the O(1)ÐO(1) and O(2)ÐO(2) edges and megahertzs). This substance should also possess con- form a three-dimensional framework where each LuO8- siderable splitting of levels of paramagnetic impurity polyhedron is surrounded by its four nearest neigh- (of the order of several dozens of gigahertzs) in the bors—similar polyhedra. The voids of the three-dimen- absence of a magnetic field. sional framework are filled with the zigzag chains of Unfortunately, we did not find any substances with NbO6-polyhedra sharing the O(2)ÐO(2) edges and, known NQR and EPR frequencies of the paramagnetic thus, forming columns. In a layer parallel to the xy dopants. Therefore, we used as a promising working plane, the LuO8- and NbO6-polyhedra share vertices in such a way that each LuO -polyhedron shares vertices material the compound LuNbO4, in which the NQR fre- 8 quency for lutetium nuclei lies in the proper range [2]. with two adjacent NbO6-polyhedra. A LuO8-polyhe- Single crystals of this compound doped with ~0.3 at. % dron shares two vertices and two edges with NbO6- Cr3+ ions were obtained, and the EPR spectra of chro- polyhedra from the neighboring layers. mium ions were studied. The calculation of the spectra The above crystals possess ferroelastic properties allowed us to determine the types of centers formed [7Ð9]; with an increase in temperature they undergo the by Cr3+ ions and the level splitting of a paramagnetic ferroelastic phase transition accompanied by an Cr3+ ion. increase in the symmetry to the tetragonal (4/mF2/m). The physical properties of the LuNbO crystal are stud- LuNbO4 crystals belong to ABO4-type compounds, 4 where A is a rare earth element and B = Nb or Ta, ied insufficiently because of the difficulties encountered in the growth of single crystals. Our study is ded- with the structure of fergusonite (Y1 – xYbxNbO4), a mineral which crystallizes in the monoclinic space icated to the growth of LuNbO4 : Cr single crystals and group I2/a [3Ð10]. The structure of this mineral is the study of the EPR spectra of Cr3+ ions introduced into their lattice. derived from the scheelite (CaWO4) structure, whose projection onto the xy plane is shown in Fig. 1 [4]. This structure has the crystallographically independent posi- GROWTH OF SINGLE CRYSTALS tions of Lu and Nb and two crystallographically independent positions of oxygen (O1 and O2). Lutetium is The crystals were grown by the floating zone tech- coordinated with oxygen atoms forming a distorted nique on an URN-2-ZP setup using optical heating eight-vertex polyhedron. In contrast to the scheelite [11]. Polycrystalline rods of the composition

EPR SPECTRA 321

(Lu0.997Cr0.003)NbO4 synthesized using the conventional ceramic technology served as the starting material for 2.34 2.34 crystallization. The rods 8 mm in diameter and 90 mm Lu 2.28 in length were ﬁred and sintered at temperatures of 2.28 1200 and 1350°C, respectively. Zone recrystallization 2.29 2.29 proceeded at a linear rate of 4.5 mm/h, with the upper 2.38 2.38 ceramic rod and the lower recrystallized ingots being 2.39 2.39 rotated in opposite directions with an angular velocity O2 of 40Ð60 rpm. 1.93 Nb 1.93 The ingots thus obtained were boules of monoclinic 1.85 1.85 LuNbO4 single crystals ~8 mm in diameter and 20 mm in length. The crystal layers up to ~1 mm in thickness O1 were transparent and had a light yellowÐgreen color and possessed a pronounced cleavage along the (010) planes. The studies of the platelike crystals with the (010) basal plane in an MMP-2 polarization microscope showed that they consist of polysynthetic twins seen as alternating light and dark bands. The X-ray diffraction studies were performed on a DRON-4 diffractometer (CuKα-radiation). It is established that the powder X-ray diffraction pattern is well consistent with the known data on LuNbO4 crystals [3Ð10] and can be indexed on the basis of the monoclinic unit-cell with the parameters a = 5.229(4) Å, b = y 10.8211(3) Å, c = 5.030(8) Å, β = 94.5(6)° (the unit- z cell parameters were calculated for the base-centered x sp. gr. C2/c often used for fergusonite crystal, a non- Fig. 1. Projection of the LuNbO4 structure onto the xy conventional version of the sp. gr. I2/a [3Ð10]). plane. The coordination polyhedra of Lu3+ and Nb5+ ions are hatched.

EPR SPECTRA Relative intensity 1.0 The EPR spectra were studied at a frequency of (a) 9.80 GHz on a Bruker ER-200tt spectrometer at room temperature. The specimen was a 1 × 3 × 6-mm single- 0.6 crystal LuNbO4 : Cr plate with the {130} basal and {010} lateral faces. The crystal was glued with differ- 0.2 ent faces to a quartz rod of the spectrometer with BF-2 glue. During measurements, the rod could be rotated Intensity, arb. units around its axis, thus changing the crystal orientation in 1.0 (b) the applied magnetic field. At an arbitrary orientation of 0.6 the crystal in the magnetic fields with an intensity of up to 9 kOe, the EPR spectrum consisted of more than 0.2 30 lines. The number, position, and intensity of the lines changed depending on the magnetic-field orienta- 1000 2000 3000 4000 5000 6000 7000 8000 tion. If the magnetic field was oriented along the mon- H, Oe oclinic b-axis, the spectrum was considerably simpli- 3+ Fig. 2. EPR spectra of Cr ions in LuNbO4 at a frequency fied (Fig. 2a). There are four mutually perpendicular of 9.80 GHz at room temperature. Vertical bars indicate the directions normal to the b-axis in the ac plane; the spec- positions of the resonance lines and their intensities; filled tra obtained along these directions are considerably circles indicate the positions of the EPR lines due to Cr3+ ions located in the lutetium sites; filled triangles indicate the simplified and are identical (Fig. 2b). In this case, the positions of the Cr3+-resonance lines in the niobium sites lines corresponding to the transitions in one center O1; inverted filled triangles, the positions of the Cr3+-reso- degenerated, and the lines corresponding to the transi- nance lines in the niobium sites O2. (a) External magnetic tions in other centers doubled, whereas within an inter- field is parallel to the C2-axis. The calculation was performed val of ~1.5°, the situation was reversed. These data at the parameter values listed in the table. (b) External magnetic field is oriented along four directions with the minimum allow one to determine the misorientation angle of the spectrum multiplicity which are normal to the C2-axis. Only symmetry axes of the local environment for different the positions of the resonance lines are calculated.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

322 BEDA et al. types of centers. The minimal linewidth on the EPR where spectrum was 10 Oe. β 2 Ð D Ð g|| Hz/2 Ð E12, 3+ a12, = ------β , The spectra observed are attributed to Cr paramag- g|| Hz Ð 2E12, netic centers. As was indicated above, the structure has β two crystallographically independent positions for Lu 2 D + 3g|| Hz/2 Ð E12, b12, = ------β ; and Nb atoms and two crystallographically indepen- g|| Hz Ð 2E12, (4) dent positions for oxygen atoms, O1 and O2. Since the β 3+ 5+ 3+ 2 D Ð 3g|| Hz/2 Ð E34, ionic radii of Cr and Nb are close [12], the Cr cat- a34, = ------, Ðg||βH Ð 2E , ions introduced into the LuNbO4 structure in low con- z 34 3+ β centrations can replace not only the Lu -positions but 2 Ð D + g|| Hz/2 Ð E34, also the Nb5+-positions. The filling of the niobium posi- b34, = ------. Ðg||βH Ð 2E , tions gives rise to an oxygen deficit either in the O1 or z 34 ψ | 〉 in the O2 position. The existence of four mutually Since the eigenstates at E = 0 are 1 3/2 , ψ | 〉 ψ | 〉 ψ | 〉 orthogonal directions perpendicular to the twofold 2 –1/2 , 3 1/2 , and 4 –3/2 , this axis C2, along which the spectra are identical, can be notation is also convenient at E ≠ 0, but one has to keep explained by the twinning characteristic of ferroelastic in mind their conventional sense. In this orientation, the crystals because of phase transition. As a result, along probabilities of transitions are determined by the these directions, one can simultaneously observe two expressions spectra formed corresponding to two mutually perpen- 2 dicular directions of the nearest environment of para- 3 3 W → ∞ a ------a + b + ------b b , magnetic center, which can be transformed into one 1/2 3/2 32 1 1 2 1 3 another by rotation through 90°. Thus, in this compound, one paramagnetic isovalent center can be 3 3 2 3+ W → ∞ a ------a + b + ------b b , formed in the Lu position and two oxygen-deficient Ð3/2 3/2 42 1 1 2 1 4 paramagnetic centers can be formed in the Nb5+ position. 2 ∞ 3 3 WÐ1/2 → 1/2 ------a2a3 + b2a3 + ------b3 , (5) The spectra were identified and the parameters of 2 2 the spin Hamiltonian were determined by comparing 3 3 2 the observed and the calculated line positions. At the W → ∞ a ------a + b + ------b b , parallel orientation (the magnetic field applied along Ð3/2 Ð 1/2 42 2 2 2 2 4 the twofold symmetry axis), the spectra were identified by both the positions and the intensities of the EPR WÐ1/2 → 3/2 ==0, W1/2→ Ð 3/2 0. lines. To describe the spectrum at the monoclinic sym- It is seen from Eqs. (3)Ð(5) that the orthorhombic metry of the nearest environment of chromium ions component of the crystal field leads to a mixing of the with S = 3/2, the following spin Hamiltonian [13] was states |3/2〉 and |–1/2〉 as well as |1/2〉 and |–3.2〉, and the used: occurrence of the additional transition |–3/2〉 |3/2〉 ≠ with W–3/2 → 3/2 0, which was forbidden earlier. Hˆ S = g||βHˆ zSˆ z + g⊥β()Hˆ xSˆ x + Hˆ ySˆ y (1) Using the above formulae, we calculated the EPR 2 2 2 spectrum in the case where the magnetic field was + DS[]ˆ z Ð SS()+ 1 + ES()ˆ x Ð Sˆ y , applied along the C2-axis. The calculated lines and the β experimental spectrum are shown in Fig. 2. Both posi- where is the Bohr magneton, g|| and g⊥ are the g-fac- tions and intensities of the lines observed in the parallel tors, and D and E are the constants that describe anisot- orientation at H = 2623, 3371, and 4360 Oe are well ropy along the monoclinic C2 axis and in the plane nor- described by the Hamiltonian at the parameter values mal to it, respectively. If the magnetic field is applied listed in table. We attribute these three lines to isovalent along the z-axis, the energy levels take the values Cr3+ centers formed by the replacement of Lu3+ by Cr3+ ions. The calculated positions and relative intensities of ()β ± ()β 2 2 the resonance lines are shown by circles in Fig. 2a. The E12, = 1/2 g|| Hz g|| Hz + D +,3E (2) lines denoted by upward-directed filled triangles (for 2 2 one center) and downward-directed filled triangles (for E , = Ð ()1/2 g||βH ± ()g||βH Ð D +,3E 34 z z another center) should be attributed to two non-equiva- 3+ 5+ and the eigenfunctions are lent Cr centers occupying the Nb positions. The parameters of the spin Hamiltonians that describe the ψ |〉 |〉 spectra of these centers are listed in the table. 12, = a12, 3/2 + b12, Ð1/2 , (3) The energy-level values and the positions of the res- ψ |〉 |〉 34, = a34, 1/2 + b34, Ð3/2 , onance lines for the perpendicular direction can be

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 EPR SPECTRA 323

3+ Parameters of EPR spectra of Cr ions in LuNbO4 niobate single crystals in experiments on the dynamic orientation of the nuclei. Type D, E, ν , g g 0 of center || ⊥ cmÐ1 cmÐ1 GHz REFERENCES Lu3+ 1.82 ± 0.01 1.90 ± 0.02 0.024 0.093 9.7 The first 1.75 ± 0.01 2.05 ± 0.02 0.17 0.14 17.7 1. V. A. Atsarkin, A. L. Barabanov, A. G. Beda, and Nb5+ ion M. V. Novitsky, Nucl. Instrum. Methods Phys. Res. A 440, 626 (2000). The second 2.2 ± 0.1 2.13 ± 0.05 0.084 0.11 13.4 5+ 2. A. F. Volkov, L. A. Ivanova, Yu. N. Venevtsev, and Nb ion L. L. Rappoport, Zh. Fiz. Khim. 56, 1002 (1982). 3. Minerals. Compound Oxides, Titanates, Niobates, Antimonates, and Hydroxides (Handbook), Ed. by obtained only numerically by solving the secular equa- F. V. Chukhrov and É. M. Bonshtedt-Kupletskaya tion corresponding to Hamiltonian (1). In this case, (Nauka, Moscow, 1967), Vol. II, issue 3, p. 248. only the g⊥-value of the g-factor was used as a fitting 4. P. A. Arsen’ev, V. B. Glushkova, A. A. Evdokimov, et al., parameter. Therefore, in Fig. 2b the line positions are Rare Earth Compounds. Zirconates, Hafnates, Niobates, ≠ ≠ indicated for Hx 0, Hy = 0 and Hx = 0, Hy 0. The agree- Tantalates, and Antimonates (Nauka, Moscow, 1985). ment between the calculated and observed spectra is sat- 5. L. A. Ivanova, E. B. Perova, and Yu. N. Venevtsev, Syn- isfactory; however, in order to describe the spectrum in thesis, Structure, and Properties of Stibiotantalites and more detail and with a higher accuracy, the measure- Fergusonites, in Series Scientific and Technological ments should also be made at another frequency. Prognosis in Catalysis, Corrosion, and Synthesis of Fer- roelectric Materials (NIITÉKhIM, Moscow, 1977). 6. X-ray Powder Data File (American Society for Testing CONCLUSIONS and Materials, Philadelphia, 1981), Card Index ASTM, File 23-1207. Thus, large LuNbO4 ferroelastic single crystals containing 0.3 at. % of Cr3+ ions are grown for the first time 7. W. Jeischko, A. W. Sleight, W. R. McCellan, and J. F. Wei- 3 her, Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. by the floating zone technique ~1 cm in volume, and Chem. B32, 1163 (1976). their EPR spectra are studied. The spectrum lines are 8. L. H. Brixner, J. F. Whitney, F. C. Zumsteg, and caused by the transitions from three paramagnetic cen- G. A. Jones, Mater. Res. Bull. 12, 17 (1977). 3+ ters formed by the replacement of an isovalent Lu and 9. A. T. Aldred, Mater. Lett. 1, 197 (1983). 5+ 3+ two nonisovalent Nb ions by Cr ions. Twinning 10. A. I. Kondrat’eva, S. K. Filatov, L. V. Andrianova, and revealed in the crystals leads to a doubling of the num- A. M. Korovkin, Izv. Akad. Nauk SSSR, Neorg. Mater. ber of lines and the formation of four principal direc- 25, 1710 (1989). tions normal to the twofold axis C2 of the crystal, with 11. A. M. Balbashov and S. K. Egorov, J. Cryst. Growth 52, the spectra along these directions being the same. The 498 (1981). spectra of these centers are described within the frame- 12. R. D. Shannon, Acta Crystallogr., Sect. A: Cryst. Phys., work of the spin Hamiltonian with S = 3/2, the D and E Diffr., Theor. Gen. Crystallogr. A32, 751 (1976). parameters ranging from 0.024 to 0.170 cmÐ1, and the 13. S. A. Al’tshuler and B. M. Kozyrev, Electron Paramag- g-factors g|| = 1.75–2.20 and g⊥ = 1.90–2.13. netic Resonance (Fizmatgiz, Moscow, 1961; Academic, The EPR frequencies of the Cr3+ ions in the zero New York, 1964). magnetic field (9.7, 13.4, and 17.7 GHz) lie in the ultra- high frequency range, which allows one to use lutetium Translated by T. Dmitrieva

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 324Ð329. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 361Ð366. Original Russian Text Copyright © 2002 by Chudakov. PHYSICAL PROPERTIES OF CRYSTALS

Comparative Analysis of Methods for Studying Internal Stresses in Crystals V. S. Chudakov Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia Received July 19, 2001

Abstract—The possibilities provided by the main polarization optical methods of studying internal stresses in differently oriented plates prepared from cubic crystals are compared. A promising method for obtaining exhaustive information on the stressed state of an object, including the trajectories and the values of the principal stresses, is described. © 2002 MAIK “Nauka/Interperiodica”.

Most of the dielectric and semiconductor cubic gence in differently oriented plates in the following crystals are transparent in the near IR range of the spec- way: for the (111)-oriented plates trum. This allows as to efficiently study their stressed ∆ 3()πσ σ ()π π states in polarized light. Such studies give valuable nn= 0 2 Ð 1 11 Ð212 + 44 /6, (1‡) information on the quality of the growth apparatus, α β shortcomings of the technologies used, and, based on 0 = ; (1b) the defect concentration, also provide a preliminary for the (100)-oriented plates certification of the crystal quality affecting the mechan- ∆ 3()σ σ ical, optical, and electrophysical properties of future n = 0.5n0 2 Ð 1 (2‡) details and devices. 1/2 ×π[]()Ð π 2 cos2 2β + π 2 sin2 2β , Internal stresses in transparent solids induce artifi- 11 12 44 α ()π π β π cial optical anisotropy and, in particular, they locally tan2 0 = 11 Ð 12 tan2 / 44; (2b) transform isotropic crystals into uniaxial (classes 43m , and for the (110)-oriented plates 432, and m3m) or biaxial (classes 23 and m3) ones [1]. ∆ 3{[ ()σπ π π ( 2 β The patterns of birefringence distribution caused by n = 0.5n0 0.5 11 Ð 12 + 44 1 cos 2 induced anisotropy and observed in polarized light are σ 2 β ()σπ π ()]2 β σ 2 β 2 essentially dependent on the orientation of the principal + 2 sin Ð 11 Ð 12 1 sin + 2 cos 2 stresses with respect to the principal symmetry axes (3‡) []π ()βσ σ 2 2 }1/2 and the direction of observation. Such a dependence + 44 2 Ð 1 sin , predetermines essential differences in the observed pat- α π ()σ σ β [ ()π π π terns in the analysis of differently oriented plates tan2 0 = 44 2 Ð 1 sin2 / 0.5 11 Ð 12 + 44 despite the same distribution of the internal stresses. ×σ()2 β σ 2 β 1 cos + 2 sin (3b) For arbitrarily oriented plates, the value of the ()σπ π ()]2 β σ 2 β induced difference in the principal refractive indices ∆n Ð 11 Ð 12 1 sin + 2 cos , (birefringence) depends on the refractive index n0 of an where α is the azimuth of the fast axis of the optical σ 0 unstressed crystal, the values of the principal stresses ij indicatrix at the point of observation measured from the and the differences between them, and the piezo-optic direction of the maximum transmission of the polarizer, coefficients π and their combinations in a rather com- β σ ij and is the azimuth of the principal stress 1 measured plicated way. In most occasions, one uses in practice from the [100] axis. thin (111)- and (100)- and sometimes also (110)-ori- The transition from the stresses in the plates to the ented plates prepared from crystals of the classes m3m volume stresses in the initial Czochralski-grown silicon and 43m . To describe the stressed state revealed with crystal is described in [2]. the aid of induced optic anisotropy, one uses the piezo- As follows from the above relationships, the exhaus- π π π optic coefficient 44 and the difference ( 11 – 12). If the tive study of a plate depends on its orientation. It is stressed state of the crystal is provided by micros- most convenient to study the (111)-oriented plates tresses, one can assume that the plane stressed state is because they provide a direct relation between the mea- ∆ α σ σ β studied for thin plates related to the induced birefrin- sured n and 0 values and the ( 2 – 1) and values.

This allows us to directly record stress diagrams by cer- allow the visual observation of the induced birefrin- tain methods. In (100)- and (111)-oriented plates, all gence. the orthogonal directions are equivalent, but the princi- The reliability of the pattern reﬂecting the stressed pal axes of the stress tensor do not coincide with state of the crystal in the polarized light depends on the α ≠ β the principal axes of the indicatrix, i.e., 0 ; the wavelength of the probing radiation, the type of polar- only exception are the points at which the principal ization, the degree of optical homogeneity of the object, stresses act along the [100]-and [110]-axes. In this and on the observation method. This is explained by the α β case, 0 = and fact that the image is an interference pattern dependent on the above factors and, therefore, is often inconsistent ∆ 3()σπ π ()σ n[]100 = n0 11 Ð 12 2 Ð 1 /2, (4) with the stress distribution. The most serious distortion of the information on ∆ 3π ()σ σ n[]110 = n0 44 2 Ð 1 /2. (5) the stress distribution is characteristic of a linear ∆ crossed polariscope in which the brightening level I In the general case, in order to determine n in the plate ∆ α of the (100)-cut, one ﬁrst has to calculate the β value. depends on both n and 0: In the plate of the (110)-cut, not all the orthogonal I ∼ τδsin2 ()/2 sin2 2α , (8) directions are equivalent, which complicates the form 0 0 of Eqs. (3a) and (3b). An analysis of these relationships where τ is the optical transmission of the object. At α ° shows that it is impossible to study the internal stresses points where 0 = 0° and 90 , the maximum darkening in the (110)-plates. An exception is the four points is observed, which is limited by the quality of the polar- located at the plate edge. At two of these points, the tan- ization elements. The set of points with the maximum gents to the plate edge are parallel to [100], whereas at darkening forms isoclines moving along the object dur- two other points they are parallel to [110]. At the edge ing its rotation. Similar darkening is also observed at points of an arbitrarily oriented plate, only the uniaxial points where Γ = λ. To identify these points, one has to stresses στ tangential to the plate edge exist. If these use radiation with a different wavelength. It follows stresses in the (110) plate coincide with the [100]-and from (9) that the inconsistency of the brightness and the 2δ [110]-directions, we have stress level is explained by the term sin 0/2. ∆ 3()σπ π In a circular polariscope with two crossed quarter- n[]100 = n0 11 Ð 12 τ/2, (6) wave plates, the distortions introduced by the azimuth ∆ 3()σπ π π of the object with respect to the polarizer are removed n[]110 = n0 11 Ð 12 + 44 τ/4. (7) because To determine the parameters of the stressed state of ∼ τδ2 () ∆ α I sin 0/2 . (9) an object, one has to know the n and 0 values, which are determined with the use of various polariscopes. In polariscopes with a rotating polarization element ∆ α The values of n and 0 in polariscopes with visual and a rotating half-wave phase plate, information on the observation and in those supplied with infrared image stress distribution is distorted much less, because the transformers are determined by the Senarmont method following expressions are valid: [2] or with the aid of various compensators [3] provid- ∼ τδ ()ω α δ I()ω sin 0 sin2 t Ð 0 , (10) ing the determination of the phase difference 0 or the Γ path difference . These parameters are related to the I()ω ∼ τδsin sin22()ωt Ð α , (11) ∆ δ π∆ λ Γ 0 0 averaged n values as follows: 0 = 2 nd/ and = ∆nd, where d is the plate thickness and λ is the wave- where ω is the circular frequency of the element rota- length of the radiation used. It should be remembered tion. For a rotating phase plate, the frequency of the that for thick plates the average ∆n value can consider- probing-radiation modulation is twice as high as that δ ably differ from the maximum and, thus, can distort the for a rotating polarization element. Also, at 0 < 30°, information on the critical values of the internal one can assume, within the admissible accuracy, that stresses. This method is inapplicable to the analysis of the amplitude of the modulated signals is proportional δ microstresses because it requires the knowledge of the to 0. For (111)-plates, this allows us to record the stress stress distributions around the defects with due regard diagrams in the automatic mode without any correc- for their shapes and dimensions. tions. If the software used takes into account the signal phase, the stress diagrams can be recorded also for the The probing radiation transmitted by a stressed δ object in scanning polariscopes with a rotating element (100)- plates and at 0 > 30°. [4] (an analyzer, a polarizer, or a phase plate) undergoes Of special interest are the points at which birefrin- amplitude-phase modulation and is recorded by a pho- gence is not observed in the plane stressed state irre- todetector. The parameters of the modulated part of the spective of the stress value. As follows from (1a) and signal contain information on both ∆n and α in the (2a), birefringence in the (111)-and (100)-plates is not 0 σ σ amplitude and the signal phase. The method of the observed at points where 1 = 2. In other plates with amplitude-phase modulation of the radiation does not nonequivalent orthogonal directions, birefringence is

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 326 CHUDAKOV ment, which allows the determination of the moment of the attainment of this polarization with a high accuracy, even if the polarization elements are of insufﬁciently L high quality. The device parameters allow the determi- 0 nation of the parameters of birefringence at any point of the object without changing its azimuthal position with O O0 δ respect to the polarizer (as is necessary in a linear Z 0 polariscope) by using special software or a pro- H grammed calculator. 2α y α P'' 2 0 To determine the relationship between the birefrin-

O Oy gence parameters of the device and the object at the moment of the zero modulated signal, we can use the δ y P Poincaré sphere method [5]. Figure 1 shows the horizontal projection of this sphere. The initial linearly polarized radiation is indicated by the point H on the δ sphere equator. In a device with the parameters y and α , the linearly polarized radiation H is transformed Ly P' y into the elliptically polarized one along the arc Ly. The Fig. 1. Illustrating the transformation of the initial linearly projection of the arc Ly onto the equatorial plane is HP. polarized radiation into circularly polarized radiation with If necessary, the ellipticity parameters can be deter- the use of a tunable crystallooptical device and an arbitrarily mined proceeding from the coordinates of the point P oriented object; H is the point on the sphere equator which [5]; however, it is not necessary for the solution of our denotes the initial linearly polarized radiation, Z is the sphere pole, P is the point which denotes the elliptically problem. After the elliptically polarized light is transmitted by an object with the parameters δ and α , it is polarized radiation at the entrance of the object, Ly and L0 0 0 are the arcs of the transformation, and HP and PZ are their transformed into circularly polarized light along the arc δ projections onto the equatorial plane of the sphere; y and L (point Z). The point Z is the sphere pole and PZ is the δ 0 0 are the phase differences introduced by the device at the α α projection of the arc L0. The solution of the problem azimuths of the fast axes y and 0, the points P' and P'' are the projections of the point P onto the planes containing the allows one to establish the relation between the parame- δ arcs Ly and L0, respectively, and OO0 and OOy are the direc- ters of the object and the device. First, 0 is determined as tions of the optical axes of the object and the device, respectively. δ ()δ α 0 = arccos sin y sin2 y . (13) δ α not observed at the points with other stress ratios. Thus, Then, using the data on 0, the azimuth 0 is calculated at points with zero birefringence in the (110)-plates, the from principal stresses act along the two- and fourfold axes α π α ()δ α δ with the stress ratios 2 0 = /2+ 2 y Ð arcsin cos y sin2 y/sin 0 . (14) σ /σ = 2()π Ð π /()π Ð π + π 1 2 11 12 11 12 44 It is assumed in Eqs. (13) and (14) that the calibrated δ α or scales of the device have zero readings of y and y. However, at initial parameters of zero, both the polari- σ σ []π ()π π 1/ 2 = 21+ 44/ 11 Ð 12 . (12) scope with a rotating element and the linear polariscope are insensitive to the stressed regions of the object in As was indicated above, the visual information on which the principal stresses are either parallel or the birefringence distribution, irrespective of the orthogonal to the direction of the maximum transmis- method of its observation, is distorted by nonuniform sion. Therefore, technologically, it is more convenient transmission. Birefringence in optically inhomoge- to use a crystalloptical device with the zero setting cor- neous objects can be quantitatively evaluated with the δ π α π responding to the conditions y = /2 and y = /4. use of compensators. However, calculations show that A device with such parameters is equivalent to a quar- it is more convenient to use a crystallooptical device ter-wave plate located along the diagonal to the direc- with a variable path difference and methods using a tion of the maximum transmission. In this setting, the rotating element, although for different purposes than device has maximum sensitivity to internal stresses in a compensator. Such a device is placed immediately irrespective of the object azimuth. The use of the scales behind a polarizer and transforms the linearly polarized with the changed zero reading is characterized by dif- radiation into elliptically polarized radiation with the δ α ferent relation between the parameters y and y and the parameters providing its transformation, upon the parameters of the object δ , and α : transmission through the object into a circularly polar- 0 0 ized radiation. Like natural radiation, the circularly δ ()δ' α' polarized radiation is not modulated by a rotating ele- 0 = arccos cos y cos2 y , (15)

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 COMPARATIVE ANALYSIS OF METHODS 327

2α = π Ð 2α' Ð arcsin()sinδ' cos2α' /sinδ , (16) plate which has a phase difference greater than 90° but 0 y y y 0 less than 180°. Let a rotating phase plate differ from a 180°-plate by the value δ'. Then, the complete spectrum δ π δ α π α where y' = /2 – y and y' = /4 – y. of the photodetector signal has three components—one The corresponding software provides the use of a constant and two modulated at different frequencies (at simple device consisting of two identical plates with a double and at a fourfold rotation frequency of the natural birefringence transforming the polarization, phase plate, respectively): e.g., thin mica sheets which can be rotated in the azi- U ∼ τ[()1 Ð sinδ sin2α sin2 ()δ'/2 muthal direction with respect to one another [6]. The 0 0 λ δ δ ω path difference of one plate should exceed /4. + cos 0 sin '2sin t (17) The characteristics of a polariscope with a rotating + sinδ cos2 ()δ'/2 sin()4ωt Ð 2α ]. phase plate studied with the aid of Muller matrices [5] 0 0 show that the inﬂuence of nonuniform transmission on To exclude the dependence of U on τ, only two mod- information about the distribution of induced birefrin- ulated components are recorded simultaneously along gence at the two-channel recording of the modulated two different channels calibrated prior to measure- signals can be eliminated. To implement the corre- ments; τ is excluded during the signal separation. The δ sponding system, it is necessary to use a rotating phase true value of 0 is calculated by the formula

δ {}[]ω ω ()δ ω ω (18) 0 = arctan 2Uk()4 U0()4 tan '/2 /Uk()2 U0()2 ,

ω ω where U0(4 ) and U0(2 ) are the signals from the stresses, at any point of which the direction of one of ω ω points of the object and Uk(4 ) and Uk(2 ) are the cal- the principal stresses coincides with the tangential ibration levels of the signals. The channels are cali- direction. In addition to the information on the direc- brated in an unusual way—in the absence of the object. tions of the principal stresses, the set of these trajecto- ω In this case, the value of Uk(4 ) is measured for a lin- ries demonstrates, similarly to a topographic map, the ω sites of stress concentration. Moreover, the trajectories early polarized radiation, whereas Uk(2 ), for a circularly polarized one. To transform the radiation, a special of the principal stresses provide the separate measure- tunable crystallooptical device is placed immediately ments of the principal stresses. after the polarizer. The linear polarization is attained at It is impossible to record all the families of the tra- δ δ α y = 0°, whereas the circular one, at y = 90° and y = jectories of principal stresses without the invocation of α 45°. The information about the azimuth 0 of the fast modern means of automation and computerization. It is axis is contained in the signal phase at the fourfold especially important for crystals. One of the most modulation frequency. Thus, modern automation meth- promising methods for recording the trajectories of the ods provide the successful implementation of the above principal stresses is the controlled motion of the stage. algorithm for studying internal stresses during the scan- If necessary, the stage motion is synchronized with a two- ning process, even for optically inhomogeneous coordinate plotter so that it can move in any direction by objects. the command of the automated system. These commands are worked out by a computer on the basis of the informa- The studies showed some other advantages of tion on the azimuth of the fast axis with the subsequent polariscopes with a rotating phase plate. Figure 2 shows determination of the azimuth of the principal stress. the curves recorded during scanning of a plate prepared Depending on the operation speed of the system, the table from a single crystal of neodymium-doped yttrium-alu- moves along the trajectories of the principal stresses either minum garnet. Curve 1 was recorded for a rotating ana- in the stepwise or in continuous modes. lyzer and curve 2, for a rotating phase plate. Earlier, using the method described in [7], we determined the A more complicated stepwise scanning of an object local inclusions in this plate with a noticeable polariz- along the trajectory of the principal stress provides the ability, especially well pronounced at the plate edges. It measurement of the principal stresses. The solution of is seen from these curves that the presence of polarizing this complicated problem is based on the Lamé-Max- defects considerably distorted the information on the well theorem [8], according to which the equation of induced birefringence in the case of a rotating analyzer. the equilibrium state at point i of the stressed object in The absorbing and the polarizing defects differently the integral form is distort the information on birefringence distribution. i The absorbing defects always reduce the birefringence σ σ []()σ σ ρ values, whereas the polarizing ones can either consid- i1 = 01 Ð ∫ 2 Ð 1 / 2 dS1, (19) erably increase or decrease these values. 0 σ σ Important information on the stressed state of an where O1 and i1 are the known and the sought stresses object is provided by the trajectories of the principal at the points O and i located on one trajectory, respec-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 328 CHUDAKOV

Fig. 2. Automatic records of induced birefringence in a plate cut out from a neodymium-doped yttrium-aluminum garnet single crystal with polarizing defects. Curve 1 is obtained at the rotating analyzer and curve 2, at the rotating phase plate providing the elimination of the effect of polarization.

σ 1 σ σ ρ tively. The difference ( 2 – 1) and the radius 2 of the orthogonal trajectory are measured at the point i. Figure 3 shows a fragment of the intersection of two trajectories S1 σ of the principal stresses with an indication of all the 2 σ S2 parameters necessary for calculating 1. The object is studied along the trajectory with the index S1. ∆ i ρ S The value of 2 is determined by the method based O' on the use of the data on the change ∆β of the azimuth σ ρ of the principal stress 1 in the vicinity of the point i on 2 σ the line coinciding with the direction of 2. At a small ∆ α step S in the measurements on both sides of S1, it is ρ 2∆β possible to put, with a high degree of accuracy, that 2 = 2∆S/∆β. The above assumptions provide the determination of the principal stresses in the form σ σ ()∆βσ σ i1 = 01 Ð i2 Ð i1 /2. (20) As was indicated above, the points where the principal stresses can readily be determined, are located at the edge of the plate. Therefore, the study of the object should be started exactly at these points. Fig. 3. Parameters used in the calculation of the principal The analysis performed shows that the solution of σ σ stresses 1 and 2. The trajectories of the principal stresses the particular and the general problems associated with S1 and S2 are intersected at the point i, O' is the point with the study of internal stresses and the use of methods the known parameters, ∆S is the scanning step, ρ is the cur- ∆β 2 with a rotating polarization element or a rotating phase vature radius of S2 at the point i, and is the change in the azimuth of the point i in the orthogonal displacement from plate are much more efﬁcient than other methods. This this point by one step. advantage manifests itself in a higher sensitivity and

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 COMPARATIVE ANALYSIS OF METHODS 329 operation speed and in the possibility of using modern 3. M. Born and E. Wolf, Principles of Optics (Pergamon, means of automation and computerization in the stud- Oxford, 1969; Nauka, Moscow, 1973). ies of crystals. Moreover, the method allows the inves- 4. V. S. Chudakov, Prib. Tekh. Éksp., No. 5, 210 (1977). tigation of various defects in crystals by considering 5. W. A. Shurcliff, Polarized Light: Production and Use their polarizabilities [8], the changes in the refractive (Harvard Univ. Press, Cambridge, 1962; Mir, Moscow, indices or the absorption power [10] using photoelec- 1965). tric apparatus conventionally used for the studies of the 6. B. N. Grechushnikov, A. I. Vislobokov, E. A. Evdi- stressed state of objects, and the differences in their shchenko, et al., Kristallograﬁya 38 (2), 55 (1993) spectral characteristics [9]. [Crystallogr. Rep. 38, 164 (1993)]. 7. V. S. Chudakov and B. N. Grechushnikov, Prib. Tekh. Éksp., No. 5, 195 (1975). REFERENCES 8. M. M. Frocht, Photoelasticity (Wiley, New York, 1941; GITTL, Moscow, 1950). 1. J. F. Nye, Physical Properties of Crystals: Their Repre- sentation by Tensors and Matrices (Clarendon, Oxford, 9. V. S. Chudakov and B. N. Grechushnikov, Opt.-Mekh. 1957; Inostrannaya Literatura, Moscow, 1960). Prom-st’, No. 7, 49 (1977). 10. V. S. Chudakov, Kristallograﬁya 46, 328 (2001) [Crys- 2. V. L. Indenbom and V. I. Nikitenko, in Stresses and Dis- tallogr. Rep. 46, 284 (2001)]. locations in Semiconductors: Collection of Articles, Ed. by M. V. Klassen-Neklyudova (Akad. Nauk SSSR, Mos- cow, 1962), p. 8. Translated by L. Man

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 330Ð334. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 367Ð371. Original Russian Text Copyright © 2002 by Verkhovskaya, Grishina, Kuznetsova, Pereshivko, Krivenko, Savel’ev. PHYSICAL PROPERTIES OF CRYSTALS

Second-Harmonic Generation in Vinylidene FluorideÐTrifluoroethylene Copolymers Doped with DonorÐAcceptor Molecules K. A. Verkhovskaya*, A. D. Grishina**, N. I. Kuznetsova*, L. Ya. Pereshivko**, T. V. Krivenko**, and V. V. Savel’ev** * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 117333 Russia ** Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskiœ pr. 31, Moscow, 117071 Russia Received June 5, 2001

Abstract—The temperature dependence of the second-harmonic generation in a vinylidene fluoride–trifluoroethylene ferroelectric copolymer doped with 5 wt % of a noncentrosymmetric 4-anilino-4'-nitroazobenzene chromophore is analyzed. The second-order susceptibility χ(2) in the ferroelectric phase of these copolymers appears to be twice as high as in the poly(methyl methacrylate) with the same chromophore content. The effect is attributed to the presence in ferroelectrics of a local electric field associated with the spontaneous polarization. The intensity of the second-harmonic generation in both ferroelectric and paraelectric phases correlates with the variation of the surface potential. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION P(ω) + P(2ω) (Fig. 1c). In this case, the radiation intensity of the second harmonic [I(Ð2ω)] is related to the The generation of the second harmonic of laser radi- polarization as [I(2ω)]0.5 ∞ P(2ω) = [χ(2)E(ω)E(ω)], ation is provided by nonlinear properties of noncen- where χ(2) is the second-order volume susceptibility. Upon trosymmetric polar media. Nonlinear optical properties switching-off of the field E0, the time of the orientational arise in polymer media involving chromophores pos- relaxation of dipoles (transition from the oriented state of sessing donorÐacceptor properties giving rise to the for- dipoles in Fig. 1b to the state shown in Fig. 1a) is deter- mation of the polar structure (the arrows in Figs. 1a and mined by the structural properties of the polymer matrix 1b indicate the direction from a donor to an acceptor). and, primarily, by the relation between the chromophore The second harmonic can be generated only in a non- dimensions and the free volume of the polymer [1]. centrosymmetric medium, where donor and acceptor molecules are distributed not randomly but have the The present study deals with generation of the sec- preferred orientation in the polymer bulk. The chro- ond harmonic of a laser radiation by ferroelectric poly- mophores are oriented in the polymer bulk by applying mers doped with a noncentrosymmetric chromophore. a potential difference U inducing a constant orienting The use of a ferroelectric polymer as a matrix can promote an increase in the polar order. field E0 = U/d, where d is the layer thickness. Figures 1a and 1b show the random distribution of the dipole Hill et al. [2] observed the generation of the second moments of chromophores in the absence of the field E0 harmonic in nonlinear molecules introduced into a fer- and upon their orientation in the field E , respectively. roelectric polyvinylidene fluoride–trifluoroethylene 0 µ The electric-field vector E(ω) of a laser radiation wave matrix (a hostÐguest composite system). The 8- m- polarizes a chromophore, i.e., induces an additional thick films contained 10 wt % of nonlinear molecules intramolecular charge transport in a donorÐacceptor (4-(4'-cyanophenylazo)-NN-bis-(methoxycarbonylme- molecule. The charge transfer from a donor to an accep- thyl)-aniline. The effect was observed at room temper- tor considerably exceeds the transfer in the opposite ature (d33 = 2.6 pm/V). A linear dependence of the coef- direction. As is shown in Fig. 1c, upon the orientation ficient d33 on the concentration of nonlinear molecules of chromophores in the field E0, the volume polariza- was established. tion P induced by the electric field E(ω) has the nonzero value. As a result of the effect of the strong field of a EXPERIMENTAL RESULTS light wave, the total induced polarization includes the AND DISCUSSION contribution that is nonlinear with respect to the field and can be represented as a sum of the Fourier compo- The ferroelectric matrix used consisted of a nents of the frequencies of ω, 2ω, etc., i.e., P = P(0) + vinylidene fluoride–trifluoroethylene copolymer

SECOND-HARMONIC GENERATION 331

(a) (b) x (c) x E(ω)

P(x)

P(ω)

P(0)

t P(2ω)

Fig. 1. Distribution of donor–acceptor molecules (a) without and (b) with the ﬁeld E0. (c) Upper panel: the time dependence of the volume polarization P(x) induced by the electric ﬁeld E(ω) of a light wave along the x-axis. Lower panel: the Fourier components P(0), P(ω), and P(2ω) of the anharmonic P(x) function.

(PVDFÐTrFE), Ð(CH2–CF2)n–(–ëHF–CF2)mÐ, doped trode. A fraction of the beam separated by plane 8 is with 5 wt % of a noncentrosymmetric chromophore, 4- directed to a test channel containing inorganic crystal 4 anilino-4'-nitroazobenzene (DO-3 dispersion orange with nonlinear optical properties. In both channels, the dye). The DO-3 content corresponded to a concentra- second harmonic is separated by interference light fil- tion of 0.17 mol/dm3. The PVDFÐTrFE polymer films ters 5 and 5' transmitting in the range of 532 ± 10 nm. are polycrystals consisting of crystalline lamellas The second-harmonic intensities, I(2ω), are measured mixed with the amorphous phase. The structural ÐCH2Ð by digital voltmeters 7 and 7' simultaneously in both channels by photomultipliers 6 and 6'. Negatively CF2 unit of vinylidene fluoride possesses the dipole moment of 7 × 10–30 C m (near 2D) similar to conven- charged particles formed as a result of air ionization tional ferroelectrics. Ferroelectric switching in due to the corona discharge arising at the application of vinylidene fluoride and its copolymers is usually dis- a high voltage to needle 9 from voltage source 10 are cussed within the model of nucleation and growth of deposited on the surface of the polymer layer. The antiparallel domains [3, 4]. The process of switching of resulting surface potential U is measured by a ring elec- spontaneous polarization is usually associated with the trode connected with electrometer 11. The sample tem- rotation of a polymer link about of the molecular chain perature was varied with the use of thermal blower 12. and the propagation of such a defect (kink) along this As was mentioned above, the square root of the sec- chain. The films were obtained using copolymer grains ond-harmonic intensity, I0.5(2ω), is proportional to the with a relative content of vinylidene fluoride–trifluoro- second-order volume susceptibility χ(2) dependent on ethylene of 70Ð30 and 60Ð40 mol % (Atochem, the density N of chromophore molecules, their orienta- France). According to Furukawa [5], the heated 70Ð30 and 60–40 films undergo the transition from the ferro- ° 7' 6' electric to the paraelectric phase at T1 = 110 and 90 C, 10 respectively. The films of such compositions are char- 5' acterized by a pronounced temperature hysteresis corresponding to a first-order phase transition. On cooling 4 9 from the paraelectric phase, the phase transition to the 532 nm ferroelectric state in these two films occurs at the tem- 1 6 ° 8 peratures T0 = 75 and 70 C, respectively. 1064 nm 5 2 Figure 2 shows the block diagram of a setup for 7 measuring second-harmonic generation where 1 is an 11 3 Nd:YAG laser with the fundamental wavelength λ = 1064 nm, power of 50 mJ per pulse, and a pulse dura- 12 tion of 10 ns. A laser beam passes through the polymer layer 2 obtained by solution casting onto a glass sub- Fig. 2. Block diagram of the setup for measurements (for strate 3 with a deposited transparent In2O3/SnO2 elec- notation see the text).

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 332 VERKHOVSKAYA et al.

0.5 0.5 the switching on of the corona discharge, the generation I , mV / U U0 of the second harmonic in the sample is weak, I0.5(2ω) ~ 75 1 8 mV0.5. We examined polydomain samples. The effect of the second-harmonic generation in an unpolarized 0.75 sample is associated with a finite number of domains in 50 the studied sample region. The switching on of the corona discharge resulted in an increase in I0.5(2ω) up 0.5 to 74 mV0.5 (Fig. 3). The surface potential was found to 25 be U0 ~ 40 V at room temperature, which corresponds 0.25 ≈ × 6 to the field E0 = U0/d 9 10 V/m for the layer thickness d = 4.5 µm. According to [7], the volume suscepti- 0 bility is χ(2) (= 2d ) = 11.6 pm/V at the DO-3 density 0 100 200 300 33 t, s N = 2.3 × 1020 cmÐ3 (C = 0.38 mol/dm3) in poly(methyl methacrylate). In poly(methyl methacrylate) with the Fig. 3. The time dependences of I0.5(2ω) (open triangles, same DO-3 concentration, a signal of I0.5(2ω) = left-hand ordinate axis) and U/U0 (solid line, right-hand 0.5 ordinate axis) in an as-prepared PVDFÐTrFE (60Ð40) sam- 72 mV was detected. ple containing 5 wt % of DO-3 chromophore at room tem- The DO-3 chromophore was introduced at a con- perature. The up and down arrows indicate the instants of 3 × 20 Ð3 corona discharge switching on and off, respectively. centration of 0.17 mol/dm or at N = 1 10 cm . Therefore, according to Eq. (1), one would expect that I0.5(2ω) = 32.2 mV0.5, which corresponds to the suscep- tion, and the second-order susceptibility (polarizabil- tibility χ(2) of about 5.2 pm/V. However, as is seen from ity) β of an individual chromophore: Fig. 3, in a ferroelectric with the DO-3 concentration of 3 0.5 ω 0.5 χ(2) 0.5 ()2 3 0.17 mol/dm I (2 ) = 74 mV , and, hence, = I ()χ2ω ∝∝Nβθ〈〉cos , (1) 12 pm/V, which is twice as high as the value measured in a poly(methyl methacrylate) matrix. where N = CNA, C is the concentration measured in 3 θ The measurements made under the conditions of a mol/dm , NA is the Avogadro number, and is the angle µ corona discharge indicate weak second-harmonic gen- formed by the dipole moment 0 of the ground state of eration (I0.5 ≈ 8 mV0.5) in PVDFÐTrFE copolymers the chromophore and the constant orienting field E0. The mean value 〈cos3θ〉 is determined in terms of the without DO-3. A comparable effect was observed ear- Langevin function, lier in polyvinylidene fluorine ferroelectric polymers [2, 8]. 3 〈〉µθ (2) cos = 0 f 0E0/5kT, (2) The excessive χ value in ferroelectric polymers in comparison with the value measured in a poly(methyl where f0E0 = F0 is the local internal electric field, f0 = methacrylate) matrix may be interpreted as follows. ε(n2 + 2)/(2ε + n2), where ε is the dielectric constant and Polarization provides the alignment of the dipoles of a ≈ n is the refractive index [f0 1.44 for the poly(methyl PVDFÐTrFE copolymer in crystalline regions along the methacrylate)]. For the ferroelectric PVDFÐTrFE (70Ð film normal. In this case, the dye molecules are also 30) polymer, the local electric field provided by sponta- reoriented. The investigation of the Stark effect for dye neous polarization is estimated as [6] molecules introduced into a ferroelectric matrix [9] provided the estimation of both the polar order param- ()ε × 9 Eloc, theor ==Pind + Pspont /3 0 2.5 10 V/m, (3) eter of dye molecules for a polarized polymer and the electric-field intensity E at the site of location of dye where P and P are the induced and spontaneous loc ind spont molecules. The static electric field stabilizing the induced polarizations, respectively. For the 70Ð30 copolymer, polar order attains the value of E = 3 × 108 V/m, i.e., we have P Ӷ P = 0.065 C/m2. loc ind spont × 7 is more intense than f0E0 = 1.3 10 V/m for A strong local field in a ferroelectric partly orients poly(methyl methacrylate) by a factor of 20 [see the dipole molecules of the dyes. A resulting polar χ(2) 〈 θ〉 Eq. (2)]. This can explain a much higher value in order can be specified by the quantities cos and comparison with the value measured for a poly(methyl 〈 3θ〉 cos proportional to Eloc given by Eq. (3). In accor- methacrylate) matrix. The above E value is less by an dance with Eq. (3), the effect of the local field in the loc ≠ order of magnitude than the local-field intensity calcu- paraelectric (Pspont = 0) and ferroelectric (Pspont 0) lated theoretically according to the Lorentz model (3). phases is different. This seems to be associated with the fact that the aver- Figure 3 shows the appearance and the decrease in aging in the experiment made by Blinov et al. [9] was the second harmonic measured in an as-prepared performed for both dye molecules in ferroelectric PVDFÐTrFE (60Ð40) sample containing DO-3 upon domains differently oriented with respect to the film switching on and off of the corona discharge. Prior to normal and in the amorphous phase of the polymer.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 SECOND-HARMONIC GENERATION 333

Figure 3 shows that, upon switching-off of the corona ln(U0, V) discharge, the relaxation curve for the second harmonic 4 coincides with the decreasing curve of the surface potential. The temperature dependence of the surface potential 2 U0 during corona discharge measured as a function of a decreasing temperature of a PVDFÐTrFE (70Ð30) sample, which was in the paraelectric phase at T = 120°C, is shown in Fig. 4. It is seen that the curve abruptly ° 0 changes at 73 C, the temperature close to the tempera- 2.9 3.1 3.3 ture of the phase transition from the paraelectric to the 1000/T, K–1 ferroelectric phase. This change is caused by a decrease in conductivity σ of the polymer with the lowering of –2 the temperature in this range. Indeed, the potential U0 is related to σ as [1, 10] σ U0 = Vc(Vc – V0)/[Vc + /A], (4) Fig. 4. The temperature dependence of the surface potential where the potential Vc applied to needle 9 (Fig. 2) is during corona discharge measured during lowering of the temperature of the PVDFÐTrFE (70Ð30) copolymer with constant in all the measurements, V0 is the threshold potential for the corona discharge in air, and A is a con- 5 wt % of the DO-3 chromophore. stant. It follows from Eq. (4) that U0 is almost independent of σ in polymers with low conductivity; i.e., at I0.5, mV0.5 T, °C ӷ σ Ӷ σ 1 /(AVc). At the same time, if 1 /(AVc) in poly- a σ mer matrices, the -dependence of U0 can be repre- σ 1 120 sented as U0 = B/ , where, in accordance with Eq. (4), B = AVc(Vc – V0). Since conductivity depends on tem- σ σ ∆ ∆ perature as = 0exp(– E/kT), where E is the activation energy and k is the Boltzmann constant, we have ∆ 0.5 ln(U0) = const + E/kT. 60 The activation energy for the paraelectric phase of the copolymer estimated from the data in Fig. 4 is ∆E = 4.1 eV. Thus, the kink in the temperature dependence of the surface potential U and its dramatic increase with 0 0 the lowering of the temperature in the paraelectric 0 200 400 600 800 1000 phase also indicates the corresponding decrease in the t, s sample conductivity. Figure 5 shows the time dependence of I0.5(2ω) in an Fig. 5. The time dependence of I0.5(2ω) (open triangles, as-prepared PVDFÐTrFE (70Ð30) sample with 5 wt % of left-hand ordinate axis) and (dashed line, right-hand ordinate axis) temperature of an as-prepared PVDFÐTrFE (70Ð the DO-3 chromophore obtained under switching on 30) sample containing 5 wt % of the DO-3 chromophore. and off of the corona discharge and the varying sample The up and down arrows indicate the instants of corona dis- temperature. The range a between solid vertical straight charge switching on and off, respectively. The solid vertical lines corresponds to the paraelectric state of the poly- straight lines indicate the range a corresponding to the temperature range 110Ð75°C, within which the polymer is in mer. It is seen that the weak second harmonic is the paraelectric phase. observed in the absence of an orienting field at the beginning of the measurements at room temperature and also in the PVDFÐTrFE (60Ð40) copolymer (see shown that the second-harmonic intensity at 120°C is Fig. 3); then, its intensity increases up to the maximum lower than at room temperature. When the corona dis- value immediately after the switching on of the corona charge is switched on once again during cooling in the discharge, and slowly decreases after its switching off. range of the transition to the ferroelectric state, the sec- With an increase in the temperature, the second har- ond-harmonic intensity, first, reaches a value somewhat monic drops if the corona discharge is switched off at the temperature of the copolymer transition from the lower than the maximum one and, then, continues ferroelectric to the paraelectric phase (about 110°C). It increasing gradually and, finally, upon the transition of is seen that switching on the corona discharge at 120°C the copolymer into the ferroelectric state, returns to the 0.5 (paraelectric phase) increases the intensity of the sec- initial value; i.e., (I/I0) = 1. The above changes in the ond harmonic, whereas its switching off abruptly second harmonic correspond to an increase in the sur- decreases it (compare with the slow drop at 20°C). It is face potential with the lowering of the temperature in

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 334 VERKHOVSKAYA et al. the range of the paraelectric phase in Fig. 4. Thus, sim- 01-02-16081, and by the International Science and ilar to the case of the ferroelectric phase (Fig. 3), the Technology Center, project nos. 872-98 and PP-2207. change in the second-harmonic intensity in the paraelectric phase correlates with the varying surface potential. Figure 5 also shows that, upon the restoration REFERENCES of the ferroelectric phase after switching off the corona 1. A. D. Grishina, L. Ya. Pereshivko, T. V. Krivenko, et al., discharge at t = 600 s, the generation of the second har- Polymer 42 (11), 4837 (2001). monic is retained; i.e., the copolymer remains to be 2. J. R. Hill, P. L. Dunn, G. J. Davies, et al., Electron. Lett. polarized whereas chromophores preserve their ori- 23 (13), 700 (1987). ented state. 3. T. Furukawa, M. Date, and G. E. Johnson, J. Appl. Phys. 54, 1540 (1983). 4. T. Furukawa and G. E. Johnson, Appl. Phys. Lett. 38 CONCLUSIONS (12), 1027 (1981). 5. T. Furukawa, Phase Transit. 18, 143 (1989). Thus, in a ferroelectric PVDFÐTrFE copolymer 6. L. M. Blinov, K. A. Verkhovskaya, S. P. Palto, et al., Kri- doped with a noncentrosymmetric chromophore (the stallograﬁya 41, 328 (1996) [Crystallogr. Rep. 41, 310 DO-3 dye), the second-order susceptibility χ(2) is twice (1996)]. as high as in poly(methyl methacrylate) at the same 7. D. M. Burland, R. D. Miller, and C. A. Walsh, Chem. chromophore concentration. The effect is explained by Rev. 94 (1), 31 (1994). a considerably more intense local orienting electric 8. A. Wicker, B. Berge, T. Lajzerowicz, and J. F. Legrand, ﬁeld associated with spontaneous polarization of the J. Appl. Phys. 66 (1), 342 (1989). ferroelectric. 9. L. M. Blinov, K. A. Verkhovskaya, S. P. Palto, et al., Izv. Akad. Nauk, Ser. Fiz. 60 (10), 171 (1996). 10. M. Scharfe, Electrophotography Principles and Optimi- ACKNOWLEDGMENTS zation (Wiley, New York, 1984). This study was supported by the Russian Founda- tion for Basic Research, project nos. 99-03-32111 and Translated by R. Tyapaev

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 335Ð339. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 372Ð376. Original Russian Text Copyright © 2002 by Bush, Shkuratov, Kuz’menko, Tishchenko.

CRYSTAL GROWTH

Growth and Morphological Study of Copper Oxide Single Crystals A. A. Bush*, V. Ya. Shkuratov*, A. B. Kuz’menko**, and E. A. Tishchenko** * Moscow State Institute of Radioengineering, Electronics, and Automation (Technical University), pr. Vernadskogo 78, Moscow, 117454 Russia e-mail: [email protected] ** Kapitsa Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 117973 Russia Received February 21, 2000; in ﬁnal form, July 27, 2001

Abstract—Experiments on the growth of CuO single crystals by crystallization from ﬂux in the CuO–Bi2O3Ð PbOÐPbF2, CuOÐBi2O3ÐLi2O, CuOÐBi2O3ÐB2O3, CuOÐBaOÐY2O3, and CuO–MOx systems (M = P, V, or Mo) have been performed. The best results were obtained in crystallization in the CuOÐBi2O3ÐPbF2 system: prismatic single crystals of platelet- and needlelike or isometric habit with dimensions up to 1 × 10 × 10, 1 × 1 × 20, or 6 × 6 × 8 mm, respectively, have been grown. The CuO crystals show polysynthetic twinning in the form of numerous alternating light and dark bands bound by systems of parallel straight lines on the {110} and {111} faces. A possible model of twinning associated with the Cu2O CuO transformation is considered. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION CRYSTAL GROWTH

Recently, copper oxide (II) (CuO) crystals correspond- Since the reversible CuO Cu2O phase transition ing to the natural mineral tenorite [1] have attracted occurs at 1026°C [18], it is very difficult to grow CuO considerable attention because of their unusual mag- crystals from the native melt under normal pressure. netic properties and possible high-Tc superconductivity Therefore, we grew these crystals by crystallization (HTSC) [2Ð9]. Copper oxide exhibits the properties of from fluxes of various compositions. low-dimensional antiferromagnets (AFM) with two When choosing the starting compositions for crys- points of magnetic transitions: the transition from the tallization, we took into account the characteristic fea- paramagnetic phase to AFM with incommensurate tures of the phase diagrams of the CuO systems with modulations of the magnetic structure at TN1 = 230(2) K Bi2O3, SrO, BaO, YBa2Cu3Oy, R2O3 (R = Y, Eu, or Dy), and the transition to the commensurate AFM phase at etc. [14, 15, 18Ð23] and the other known data on crystal TN2 = 213(1) K. The temperature dependence of the growth of CuO. These data showed that it is expedient magnetic susceptibility shows small jumps along the b- to use the Bi2O3 flux. However, these data are incom- and c-axes at TN2. Above TN1, the susceptibility contin- ° plete and do not provide unique information on the ues increasing to form a broad maximum at ~550 C compositions of the melts. Therefore, to choose the rather than decreases according to the CurieÐWeiss law, optimum systems, we had to perform the trial experi- which indicates the preservation of the magnetic short- ments on crystallization in the CuOÐBi2O3, CuOÐ range order above TN1. The magnetic properties of CuO Bi2O3–MOx (M = PbO, PbF2, Li2O, or B2O3), CuOÐ are similar to those of the YBa2Cu3O6 HTSC phase PbO, CuOÐPbOÐPbF2, CuO–MOx (M = Li, P, V, or [4, 9]. Moreover, CuO is a component of almost all Mo), and CuOÐBaOÐY O systems. known mixed-oxide compounds exhibiting HTSC 2 3 associated with the existence of CuÐO planes [9]. Thus, The initial components were oxides, carbonates, the comprehensive study of the structure and the prop- and fluorides of the corresponding metals of the analyt- erties of CuO are of interest and importance for clarify- ical-purity grade. The acetone-homogenated mixtures ing the nature of high-temperature superconductivity. of the starting components were heated in alundum crucibles up to temperatures exceeding their melting The known CuO crystals [3, 4, 10Ð17] do not fully ° satisfy the requirements of size and quality, which hin- points (~1060 C), were kept at the maximum temperatures for 1–3 h, and then cooled, first, to subsolidus ders their further study. In this connection, the studies ° aimed at synthesis of large high-quality copper oxide temperatures (~600 C) at a rate of 5 K/h, and then to single crystals are very important. Below, we describe room temperature in the switched-off furnace. the synthesis of CuO single crystals and their X-ray dif- The best results were obtained upon crystallization fraction and morphological studies. of the following compositions: CuO 0.8Ð0.7; Bi2O3

336 BUSH et al.

(a) (b) (c)

(111) (001)

c b (111) c c b bb aa (111) (110) (110) a aa

(110)

Fig. 1. Habit of (a) needle-like, (b) platelet-like, and (c) isometric CuO crystals (the horizontal and the diagonal hatchings indicate the polysynthetic twins in systems 1 and 2, respectively; a, b, and c indicate the orientations of the crystallographic axes).

0.2Ð0.3; and xPbF2 (x = 0Ð0.30). The surfaces of the plane. In the different parts of the crystal, the distances crystallized melts had platelet-like CuO crystals with a between the neighboring bands vary from several 10 × 20-mm basal plane, and the plate thickness was up microns to several millimeters. In most cases, the width to 1 mm. These crystals contained single crystals of of light bands considerably exceeds the width of dark CuO in a needle-like form (elongated in the c-axis) and ones. These bands are also clearly seen in the reflected 1 × 1 × 10 and 6 × 6 × 8 mm isometric prisms (Fig. 1). light in the optical and metallurgical microscopes. Black crystals exhibited cleavage along the (110) and These bands are also visible to the naked eye when (001) planes and had the characteristic {110}, {111}, studying the natural faces at different angles. Investiga- and {001} growth forms (Fig. 1). The crystals also tion of the crystals during heating demonstrates that the showed polysynthetic twinning which manifested itself above-described twin structure remained practically in the presence of alternating light and dark bands on unchanged up to ~1000°C. the crystal faces. The angles between the crystal faces were measured Another characteristic defect of CuO crystals are the on an RGNS-2 goniometer. The angle between the various inclusions. The surfaces of natural and polished (110) planes of the adjacent regions of the faces sepa- ° faces had Bi2O3, PbO, and Al2O3 inclusions inherited rated by the straight lines equals 11.2 . This fact and the from the flux and the remaining polishing paste. The X-ray diffraction patterns measured from different dimension of these inclusions attains a value of 0.5 mm. crystal faces indicate that the twins of system 1 are The X-ray diffraction pattern of powdered crystals adjacent along the (001) planes. The adjacent twins are characterized by the parallel orientation of their a- and obtained on a DRON-4 diffractometer (filtered λCuKα radiation) corresponds to the ASTM data for CuO [24]. b-axes, whereas the unit cell changes its orientation in such a way that the angle between the c-axis and the Based on these data and the results obtained in [1, 4, 5, ° ° 8, 11, 16, 17, 23, 25, and 26] we indexed the X-ray pat- (001) plane decreases from 99.5 to 80.5 when cross- tern within the monoclinic unit cell with the parameters ing the twin boundary [the (001) plane] (Fig. 3). Thus, a = 4.688(5) Å, b = 3.420(5) Å, c = 5.132(3) Å, β = the crystal structures of the adjacent twins in system 1 99.5(1)°. An examination of platelet-like CuO crystals are related by the mirror-reflection (001) plane. revealed no pyroelectric effect in the temperature range An examination of some crystals revealed, along of 80Ð300 K, which indicated the absence of phase with the above-described system of bands on the {110} transitions with the lowering of the symmetry down to faces, analogous systems of straight bands which inter- polar in this temperature range. sect the first system at a certain angle. The most common system is system 2 (Fig. 2). The bands of system 2 form an angle of 120° with the bands of system 1 in the POLYSYNTHETIC TWINNING (010) plane. The existence of various band systems Polysynthetic twins were observed on the {110} and shows the complex character of CuO twinning. {111} faces in the form of numerous alternating light Twinning in natural CuO minerals was first reported and dark bands limited by the systems of parallel in [1]. Polysynthetic twinning of CuO microcrystals straight lines (Figs. 1Ð3). In the most widespread sys- was observed in [27Ð29]. Apparently, the twin forma- tem 1 (Fig. 2), these bands are parallel to the (001) tion is not associated with the growth conditions of

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 GROWTH AND MORPHOLOGICAL STUDY 337

‡ ‡ 2

1 2

(a) 10 µm (b) 100 µm

Fig. 2. Polysynthetic twinning observed on the {010} faces of CuO crystals in a metallurgical microscope; 1 and 2 indicate two different systems of polysynthetic twins.

CuO single crystals. Twinning is inherent in the CuO (001) phase. Evidently, twinning of CuO crystals produces a strong effect on their physical properties and, therefore, should be taken into account when interpreting the crystal properties. c c2 (110)2 POSSIBLE TWINNING MODELS Usually, polysynthetic twinning is associated with (110) phase transitions of the distorting (non-reconstructive) c 1 c1 type [30, 31]. On cooling a high-temperature high-sym- (110) metry (prototype) phase, the cooperative atomic dis- 2 placements occur at the phase-transition point. These displacements can be differently oriented with respect to the initial crystal lattice in different regions of the crystal and can give rise to polysynthetic twinning in c (110)1 2 the crystals in the low-temperature phase. The local (110)2 regions of the crystal, which undergo different transformations, become twins with respect to each other. The measurements of the temperature dependence c (5.13 Å) (110)1 of the unit-cell parameters showed that CuO undergoes (110)2 1 no phase transitions in the temperature range of 20Ð ° 1000 C [25]. The CuO Cu2O transformation at 1026°C accompanied by the change in the symmetry (C c Pn m) is the only known transformation b Å 2/ 3 b (3.42 ) taking place during the heating of CuO crystals [1, 8, 18]. β = 99.5° This transformation is of a reconstructive nature and is accompanied by considerable changes in the unit-cell (110)1 parameters (a = b = c = 4.26 Å a = 4.69 Å, b = 3.42 Å, c = 5.13 Å; α = β = γ = 90° α = γ = 90°, Å β = 99.5°) and by chemical reactions of oxygen evolu- a (4.69 ) tionÐabsorption. Apparently, this transformation should result in crystal destruction, and, hence, at ﬁrst glance, Cu2O cannot be the prototype phase for CuO. Fig. 3. Schematic representation of polysynthetic twinning Possibly, the CuO crystals are related to the proto- along the (001) planes of a CuO crystal; the angles between type phase with a high temperature of transition to the the (110)1 and (110)2, (110)2 and (110)1, (001) and (110)1, prototype phase and which decompose prior to the (001) and (110)2, and (110) and (1 10) faces are 11.3°, attainment of this temperature. In these cases, the hypo- 168.7°, 84.4°, 95.6°, and 73.1°, respectively.

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 338 BUSH et al.

DTA (‡) (b)

1000 T T , °C T 500 1020°C DTA 0 50 100 150 200 50 100 150 200 250 300 t, min t, min 1010°C 0 CuO(m = 1116.8 mg) DTG CuO (m = 1045.8 mg) DTG(dm/dt)

, mg –50 TG(m)

m m = 112.3 mg

∆ TG 1050°C

–100 Cu2O 1040°C 1000°C

1050°C 1120°C

Fig. 4. Derivatograms of a high-purity grade CuO powder measured in air at the maximum heating temperatures (a) 1060°C and (b) 1200°C; T, TG, DTA, and DTG are the curves of thermal, thermal gravimetric, differential thermal, and differential-thermal gravimetric analyses, respectively; m is the initial weight of the specimen. thetical symmetry of the prototype phase should be complex kinetics of this transformation and the exist- determined by the symmetry of the low-symmetry ence of a number of intermediate Cu 2+ Cu 1+O phase and by the orientation of the domain structure of 12Ð x 2x 1 Ð x the latter. In the case of CuO, the untwinning of states. Derivatograms obtained from CuO on a PaulikÐ polysynthetic twins of system 1 is accompanied by an Erdey derivatograph (Fig. 4) show that the Cu2O increase in the crystal symmetry from monoclinic to at CuO transformation occurs within a long temporal and least orthorhombic (β 90°). The presence of other temperature ranges sufﬁcient for the crystal lattice to be systems of polysynthetic twinning in the CuO crystals rearranged without destruction of the crystal. indicates a higher symmetry of the prototype phase. According to [31], the symmetry change from Pn3m These results, along with the analysis of the CuO to C2/c in the transition from Cu2O to CuO indicates behavior during heating and crystallization (Fig. 4) that CuO belongs to the m3mF2/m class of ferroelastics [1, 8, 14, 18, 19, 21, 23, 25, 28], give grounds to con- characterized by 12 possible orientational states of sider the cubic Cu2O phase as the prototype high-sym- domains and a large number of various planes that can metry phase of CuO crystals. According to [28], the play the role of domain boundaries [32]. thermal treatment of CuO single crystals under the moderately reductive conditions at 400°C leads to the evolution of oxygen from CuO and its transformation ACKNOWLEDGMENTS into Cu2O accompanied by the disappearance of twin- We are grateful to H. Bron (Materials Science ning. The crystal structures of CuO and Cu2O are char- Group of the Department of Applied Physics at the Uni- acterized by different distortions of the crystal lattice versity of Groningen, the Netherlands) for his help in and positions of the oxygen atoms; however, the copper processing the crystal surfaces and obtaining their atoms occupy analogous positions in these crystals [1, images. 8, 18]. It should also be noted that polysynthetic twinning appearing in the phase transformation, which is This study was supported by the Russian Founda- accompanied by a change in the oxygen content in the tion for Basic Research, project no. 99-02-17752. unit cell, was observed in the crystals of the Y-123 HTSC phase [29]. REFERENCES Apparently, the ability of the crystal lattice to rear- 1. Minerals. Simple Oxides. Handbook, Ed. by range without destruction of the crystal in the F. V. Chukhrov (Nauka, Moscow, 1965), Vol. II, issue 2, Cu2O CuO transformation is associated with the p. 41.

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2. B. X. Yang, J. M. Tranquda, G. Shirane, et al., Phys. Rev. 17. X. G. Zheng, M. Suzuki, and C. N. Xu, Mater. Res. Bull. B 38 (1), 174 (1988). 33 (4), 695 (1998). 3. O. Rondo, M. Ono, E. Sugiura, et al., J. Phys. Soc. Jpn. 18. Minerals. Handbook, Ed. by F. V. Chukhrov et al. 57 (10), 3293 (1988). (Nauka, Moscow, 1974), nos. 1, 2. 4. T. A. Arbuzova, A. A. Samokhvalov, I. B. Smolyak, 19. A. A. Zhokhov, Zh. D. Sokolovskaya, G. K. Baranova, et al., Pis’ma Zh. Éksp. Teor. Fiz. 50 (1), 29 (1989) and I. I. Zver’kova, Sverkhprovodimost: Fiz., Khim., [JETP Lett. 50, 34 (1989)]. Tekh. 3 (12), 161 (1990). 5. T. A. Arbuzova, A. A. Samorokov, I. B. Smolyak, et al., 20. J. Sestak, J. Therm. Anal. 36, 1639 (1990). J. Magn. Magn. Mater. 95, 168 (1991). 21. M. P. Kulakov and D. Ya. Lenchinko, Thermochim. Acta 6. U. Kobler and T. Chattopadhyay, Z. Phys. B 82 (3), 383 188, 129 (1991). (1991). 22. B.-J. Lee and D. N. Lee, J. Am. Ceram. Soc. 74 (1), 78 7. P. J. Brown, T. Chattopadhyay, J. B. Forsyth, et al., (1991). J. Phys.: Condens. Matter 3, 4281 (1991). 23. A. A. Fotiev, B. V. Slobodin, and V. A. Fotiev, Chemistry 8. E. Gmelin, Indian J. Pure Appl. Phys. 30, 596 (1992). and Technology of High-Temperature Superconductors 9. Chemistry of High-Temperature Superconductors, Ed. (Ural’skoe Otd. Ross. Akad. Nauk, Yekaterinburg, by D. L. Nelson, W. S. Whittingham, and Th. F. George 1994). (Am. Chemical Society, Washington, 1987; Mir, Mos- 24. ASTM, ﬁle No. 5-661. cow, 1988). 10. B. M. Wanklyn and D. J. Garrard, J. Mater. Sci. Lett. 2, 25. M. I. Domnina, S. K. Filatov, I. I. Zyuzyukina, and 285 (1983). L. P. Vergasova, Izv. Akad. Nauk SSSR, Neorg. Mater. 22 (12), 1992 (1986). 11. A. A. Samokhvalov, N. N. Loshkareva, Yu. P. Sukho- rukov, et al., Pis’ma Zh. Éksp. Teor. Fiz. 49 (8), 456 26. J. B. Forsyth and S. Hull, J. Phys.: Condens. Matter 3, (1989) [JETP Lett. 49, 523 (1989)]. 5257 (1991). 12. B. A. Gizhevskiœ, A. A. Samokhvalov, N. M. Chebotaev, 27. G. N. Kryukova, V. I. Zaikovskii, V. A. Sadykov, et al., et al., Sverkhprovodimost: Fiz., Khim., Tekh. 4 (4), 827 J. Solid State Chem. 73, 191 (1988). (1991). 28. V. A. Sadykov, S. F. Tikhov, G. N. Kryukova, et al., 13. Yu. S. Ponosov, G. A. Bolotin, N. M. Chebotaev, et al., J. Solid State Chem. 73, 200 (1988). Sverkhprovodimost: Fiz., Khim., Tekh. 4 (7), 1422 29. H. Schmid, E. Burkhardt, E. Brixel, et al., Z. Phys. C 72, (1991). 305 (1988). 14. C. Chen, Y. Hu, B. M. Wanklyn, et al., J. Mater. Sci. 28, 30. R. W. Cahn, Adv. Phys. 3 (12), 363 (1954). 5045 (1993). 31. K. Aizu, J. Phys. Soc. Jpn. 27 (2), 387 (1969). 15. C. Chen, Y. Hu, B. M. Wanklyn, et al., J. Cryst. Growth 32. J. Sapriel, Phys. Rev. B 12, 5128 (1975). 129, 239 (1993). 16. T. Ito, H. Yamaguchi, K. Okabe, and T. Masumi, J. Mater. Sci. 33, 3555 (1998). Translated by T. Safonova

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

Crystallography Reports, Vol. 47, No. 2, 2002, pp. 340Ð344. Translated from Kristallograﬁya, Vol. 47, No. 2, 2002, pp. 377Ð381. Original Russian Text Copyright © 2002 by Konstantinova, Lozovskiœ. CRYSTAL GROWTH

Effect of Growth Conditions on Formation of Grain Boundaries in Epitaxial Silicon Layers G. S. Konstantinova and V. N. Lozovskiœ Novocherkassk State Technical University, ul. Prosveshcheniya 132, Novocherkassk, 346400 Russia Received January 20, 2000

Abstract—The formation of grain boundaries of the general type, along with small- and large-angle symmetric grain boundaries with the 〈110〉 axis in the epitaxial layers grown onto bicrystal substrates by the method of thermal migration has been studied. The solvent was aluminum. It is shown that if the grain boundaries in the epitaxial layer are tilted to the crystallization front or if there is a temperature gradient tangential to this front, their orientation differs from their orientation in the substrate. The large-angle symmetric boundaries are more stable than the boundaries of the general type. The grain-boundary energy and rotation moment of large-angle symmetric boundaries are evaluated. © 2002 MAIK “Nauka/Interperiodica”.

INTRODUCTION ment of the groove arises, and a grain boundary in the The formation and development of small- and large- epitaxial layer is formed normally to the initial crystal- angle symmetric grain boundaries in epitaxial silicon lization front CF0. layers grown by the thermally-induced migration If the substrate dictates the formation of the grain method under conditions where the grain boundaries boundary at an angle α to the normal (Fig. 1b), the are normal to the crystallization front have been studied groove shape becomes asymmetric. The tangential dif- elsewhere [1]. It was shown that many large-angle 〈 〉 fusion flows can arise in the groove, which provide a boundaries with the 110 axis are split, and that the balance between the surface tension of grain boundary splitting reactions can be used to evaluate grain-bound- and the interface. The additional tangential diffusion ary energies. Below, we describe the study of grain- flow of atoms of the substance to be crystallized can boundary formation in epitaxial layers under conditions also be caused by the applied temperature gradient Gt, where the grain boundaries are tilted to the crystalliza- which is tangential to the crystallization front. The tion front and when there is a tangential temperature additional diffusion flows give rise to additional dis- gradient. The results obtained enable us to perform not placement of the groove along the crystallization front only relative evaluations, but also provide the evalua- in the direction opposite to that of the diffusion flow tion of grain-boundary energies and the rotation moments that cause the deviation of the grain boundary from its initial position toward the orientation with the (a) (b) minimum energy. x s Gt CALCULATION OF THE POSITION f OF THE GRAIN-BOUNDARY CF G G III IN THE EPITAXIAL LAYER G IIIe Let a grain boundary GB between grain 1 and grain 2 δ α l0 (Fig. 1a) in the substrate be normal to the initial crystal- CF0 B B lization front CF0. During the process of thermally- sub 12sub 1 2 induced migration, a plane liquid layer of flux is moving from the substrate through the crystal-source along the direction of the temperature gradient. The epitaxial Fig. 1. Schematic cross section of the sample illustrating the process of thermally-induces migration: (1, 2) adjacent layer is left behind the crystallization front (CF). The grains in the substrate; (e) epitaxial layer; (s) crystal-source; grain boundary intersects the crystallization front and (f) flux; (sub) substrate; (CF) the crystallization front; forms a groove, which can be treated as a liquid linear (CF0) the initial crystallization front; (B) grain boundary in inclusion on the crystallization-front surface. The melt epitaxial layer; (G) the groove on the crystallization front; (G) and (Gt) normal and tangential temperature gradients, in the groove volume is in contact with solid phases I respectively. (a) the grain boundary in the epitaxial layer is and II. If these phases are thermodynamically equiva- normal to the crystallization front; (b) the grain boundary is lent, no driving force giving rise to tangential displace- tilted to the crystallization front.

EFFECT OF GROWTH CONDITIONS 341

(the displacement mechanism is similar to the mecha- dµ RTdC F ==------N ------N nism of thermally-induced migration). As a result, the t dx C dx grain boundary in the epitaxial layer starts deviating (3) from the orientation dictated by the substrate. RTdCdT RTdC S = ------N = ------Gt ------, In our experiments, the coherent {111}Σ = 3 grain C dT dx C dT V m boundaries with a low speciﬁc surface energy σ where µ and C are the chemical potential and the solu- (~0.05 J/m2) produced no visible grooves on the crys- 1 dC tallization front. bility of silicon in the liquid phase, respectively; ------is In all the epitaxial layers irrespective of the α-value dT the slope of the liquidus line for the binary SiÐAl phase in the presence of Gt, such boundaries showed no deviations from the crystallographic orientation. Grain diagram at the process temperature, N is the number of boundaries of the general types (with high values of σ mole of silicon per groove-length unit, S is the area of and the absence of the orientation dependence of σ) the groove cross section, and Vm is the molar volume of were formed normally to the crystallization front [2]. silicon in the liquid phase. In accordance with Eqs. (2) To a number of large-angle grain boundaries with low and (3), the position of the grain boundary GB in the σ values (the so-called special grain boundaries) there presence of Gt is determined by the condition corresponds a deep minimum on the orientational σ σ σαδ()d ()αδ dependence of . These grain boundaries deviated from Ft Ð sin Ð +0.------cos Ð = (4) the initial orientation at the angle δ (see Fig. 1b). The dδ energy gain owing to a decrease in the grain-boundary area during its deviation from the normal is compensated with an increase in σ. EXPERIMENTAL RESULTS AND DISCUSSION Assuming that the grain-boundary length in the direction normal to the drawing plane in Fig. 1 equals As in [1], the process of thermally-induced migration unity, we can state that the equilibrium value of the continued for 1.0Ð1.3 h at T = 1100°ë and G = 10 K/cm angle (α – δ) is determined by the condition dE = σdl + with the use of an aluminum solvent. The preparation ldσ = 0, where E = σl is the grain-boundary energy, of bicrystal substrates is described elsewhere [1]; the scheme of the substrate and the transverse cross section l0 σ l = ------()αδ- is the grain-boundary length, and d = of the specimen with grain boundaries in the epitaxial cos Ð layer are also considered in [1]. The angles between the dσ ------dδ . Substituting l, dl, and dσ into the initial expres- grain boundary plane and the normal were measured in dδ an optical microscope an accuracy of 1.0°Ð1.5°. The sion for dE, we have experiments were repeated from three to ﬁve times for each type of substrate. We studied grain boundaries of 1 dσ tan()αδÐ = ------, (1) the general type, along with small- and large-scale sym- σ dδ metric (special) grain boundaries with the [110 ] axis. dσ where ------δ is the rotation moment applied to the surface d Grain Boundaries of the General Type 1 dσ unit of the grain boundary and (------) is the normalized Four different variants of misorientated grains were σ dδ used to prepare the substrates. In the absence of G , the dσ t rotation moment. Since neither σ nor ------are dependent grain boundaries in the epitaxial layer were normal to δ ° d the initial crystallization front. If Gt = 5 C/cm, the grain on α, the deviation δ should increase with an increase boundaries in the epitaxial layer systematically devi- α δ in . In the presence of the tangential temperature gra- ated toward Gt. The deviation angles t for four types of dient Gt, the equilibrium position of the boundary is substrates were 17°–29°, 7°–10.5°, 8°–15°, and 10°– determined by the condition 34°, respectively. According to Eq. (4), we obtain at α = 0, dσ dEÐ0, Ftdx = (2) ------= 0, and δ = –δ dδ t α δ where x = l0 tan( – ) and Ft is the thermodynamic force per groove-length unit provided by the tempera- σ Ft = ------δ- . (5) ture gradient Gt. sin t

L V dC × –3 –1 1 Σ = ------, where VCSL and VL are the unit-cell volumes of the Substituting C = 0.53, ------= 1.06 10 ä (from the CSL dT V ° coincidence-site lattice of two neighboring grains and the unit- phase diagram of the SiÐAl system at 1100 C), Vm = cell volume of the initial crystal lattice, respectively. 11 cm3/mol, and also the average value of the groove

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 342 KONSTANTINOVA, LOZOVSKIŒ

α α plane normal to the substrate (111) surface. In both cases, the grain boundaries in the epitaxial layers were [110] [111] [110] found to be normal to the initial crystallization front [111] ° [112] Gt = 0. Under the temperature gradient Gt = 5 C/cm, the grain boundaries of both types in epitaxial layers deviated from the normal toward G by angles δ = 18°–26° 1 2 t t [112] and 19°–32°, respectively. For small-angle grain boundaries, σ depends on orientation [1], therefore, σ for small-angle grain boundaries are lower than for Fig. 2. Orientations of grains intergrown during the prepa- grain boundaries of the general type. It is hardly possi- ration of the substrates and determining the formation of the ble to evaluate σ for small-angle grain boundaries ()112 Σ* = 3 grain boundary in the epitaxial layer at the because the form of the σ(δ) dependence for these angle α to the normal. boundaries is unknown.

× –12 2 cross section S = 150 10 m, we obtain Ft = 2 δ Symmetric Grain Boundaries 0.16 J/m . Substituting Ft and the average values t into Eq. (5), we obtain the speciﬁc surface energies σ = with the []110 Axis 0.41, 1.06, 0.68, and 0.45 J/m2, for the four grain The orientations of all the grain boundaries studied are boundary types respectively. These data are in good Σ agreement within an order of magnitude with the data indicated in the table as well as -values and the orienta- reported earlier [2, 3]. Because of the pronounced scat- tions of the planes normal to the grain boundaries (the ini- ter in the δ and S values, the σ-values obtained above tial crystallization front). We studied incoherent grain t boundaries, because we prepared the substrates by inter- should be considered as rough estimates. growing the grains only slightly (~1°) deviating from the misorientations indicated in the table. Grain boundaries 1 Small-Angle Boundaries and 2 do not split in epitaxial layers, whereas other grain boundaries are split with the formation of coherent We used two types of bicrystal substrates prepared by {111}Σ = 3 boundaries and other incoherent special the diffusion welding of two slightly misoriented grains boundaries [1]. In the latter case, we studied grain bound- (~1°): The grain-boundary (111) plane normal to the aries formed during splitting, e.g., {552}Σ* = 27 and substrate (110 ) surface and the grain-boundary (110 ) {221}Σ* = 9 formed by following reactions [1]:

()112 Σ*==== 3() 111 Σ 3552+27111()Σ* +3,()Σ 1 1 1T 2 (6) 0° 19.5° 0° Ð19.5° ()552 Σ*27111===()Σ 3221+9,()Σ* 1 1 1T (7) 0° Ð19.5° Ð35.3° and also grain boundary {441} Σ* = 33 formed by the and “–” indicate the counter- and the clockwise mea- reaction surements from the normal. The asterisks indicate incoherent grain boundaries, the subscripts “1,” “1T,” and ()332 1Σ* = 11() 111 1Σ = 3+() 441 1TΣ* = 33. “2” relate to grain 1, its twin [the twinning plane (111 ) ], ° ° ° (8) 1 0 Ð10 Ð35.2 and grain 2, respectively. The tilt of the grain boundary In reactions (6)Ð(8), below the symbols of grain bound- in the epitaxial layer was set during the substrate prep- aries, we also indicate the angles they form with the aration. Thus, in order to obtain grain boundary 4 in the normal to the initial crystallization front. The signs “+” epitaxial layer forming an angle α with the normal, we had to intergrow the grains shown in Fig. 2. The splitting reactions (6) at angles α = 0° and α = 19.5° are shown in The orientations of large-angle symmetric grain boundaries Fig. 3a and in Fig. 3b, respectively. The KD and KD and the planes normal to these boundaries (CF⊥) 1 2 are coherent {111}Σ = 3 boundaries, while KD is an No.123456 Σ incoherent ()552 1T * = 27 boundary. The substrate GB 111 113 221 552 441 114 structure is described in detail in [1]. The positions of the coherent grain boundaries correspond to the crystal- Σ 3 11 9 27 33 9 lographic position, whereas the incoherent grain CF⊥ 112 332 114 115 118 221 boundary deviates from it toward the normal. The reac-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 EFFECT OF GROWTH CONDITIONS 343

D D1 D D2 D1 D 2

L K M

P 2 M 1 L K C Q B C B

Fig. 3. Splitting reaction (6) on the transverse (110 ) CLKMB cross section of the substrate. The epitaxial layers start growing from the LKM surface (a) the substrate surface has the (111) orientation; splitting starts at the point K; (b) the substrate surface forms an angle of 19.5° with the (111) plane; splitting started at the points P and Q of the substrate. tions of all types resulted in the appearance in the epi- initial crystallization front, the splitting of grain bound- taxial layers of coherent {111}Σ = 3 grain boundaries ary 6 in the epitaxial layers corresponded to the following and forming no visible grooves, in accordance with the reaction: ()114 Σ*(9) (111 ) Σ = 3 + (111 ) Σ = 3. crystallographic conditions, at arbitrary angles α with 1 1 2 One of the boundaries obtained should be incoherent, the normal. The incoherent special grain boundaries since the initial grain boundary is incoherent. However, deviated toward the normal. The deviation angles δ in fact, a small-angle grain boundary was split from the increased with α and were different for different types incoherent {111}Σ* = 3 boundary. As a result, two of grain boundaries. For instance, at α = 20°, the average coherent Σ boundaries were formed in the values of the angle δ for grain boundaries 2, 3, 4, and 5 {111} = 3 epitaxial layer (which had crystallographic orientation) were 9.5°, 3.4°, 5.5°, and 6.2°, respectively (see table). In and one small-angle grain boundary forming an angle accordance with Eq. (1) for the δ values given above, the with the normal. 1 dσ values of normalized rotation moments ------are equal to σ dδ 0.19, 0.30, 0.25, and 0.26, respectively. Effect of the Tangential Temperature Gradient The temperature gradient G = 5°C/cm directed as is For the grain boundary ()221 Σ* = 9 at α = 35.3°, t shown in Fig. 1b compensated (partly or completely) σ δ 1 d for the deviation of the tilt large-angle boundaries with the deviation is = 5.2°, which yields σ------δ = d α = 5° respect to normal. For example, the {113}Σ* = 11 grain α Σ boundary at = 15° did not deviate from its position 0.58. For the grain boundary ()441 * = 33, the normal- toward the normal within the accuracy. The grain 1 dσ boundary {552}Σ* = 27 at α = 15° was formed at an ized rotation moment σ------δ = 0.26 has a much α d δ = 6° angle of = 17°. Splitting according to reaction (8) lower value than for the previous grain boundary. It proceeded under the conditions where the formed should be indicated that no splitting (7) occurs if the {441}Σ* = 33 grain boundary formed angles α = 25° α Σ α and = 35° with the normal. (As was mentioned above, grain boundary ()221 * = 9 forms an angle with the under these conditions and in the absence of G no split- |α| t normal such that > 45°. Splitting by Eq. (8) is termi- ting reaction occurred.) nated for the boundary ()441 Σ* = 33 at |α| > 25°. If the grain boundary in the epitaxial layer is normal For the grain boundary 1 (see table) at the set tilt to the crystallization front, then α = 0, and the equilib- angles in the range α = 10°–35°, the coherent {111}Σ = 3 rium condition (4) for small angles δ acquires the form: and small-angle grain boundaries were formed at σ α σδ d angles in the epitaxial layers instead of the coher- Ft Ð t Ð0.------δ = (9) ent boundaries. Small-angle boundaries were d inclined toward the normal. The deviations from the normal under the action of Σ Σ The same situation was also observed for grain Gt for the {552} * = 27, {113} * = 11, and boundary 6. For the (221) substrate plane tilted to the {221}Σ* = 9 grain boundaries were within the accu-

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 344 KONSTANTINOVA, LOZOVSKIŒ racy. Hence, the rotation moments for these boundaries can be interpreted as a variant of thermally induced dσ migration. The groove displacement results in the devi- are ------≥ F = 0.16 J/m2. dδ t ation of the grain boundary in the epitaxial layer from its initial orientation dictated by the substrate. For the {441}Σ* = 33 grain boundary, the average δ Incoherent large-angle symmetric boundaries (or deviation angle equals t = 7°. Invoking Eq. (9) and the σ special boundaries) with corresponding sharp minima 1 d on the orientational dependence of the surface energy above value σ------δ = 0.26, we obtain that, given a d α = 6° are more stable in comparison with small-angle and deviation of the grain boundary from the exact orienta- general boundaries. tion {441}Σ = 33 by an angle of 6°, its surface energy The effect of grain boundary deviation from the ori- has the value σ = 0.43 J/m2 and its rotation moment entations set by the substrate allows one to estimate dσ both the grain-boundary energy and the rotation becomes equal to ------= 0.11 J/m2. However, these val- dδ moment which “tries” to return the grain boundary to ues are only rough estimates because of the consider- the orientation with the minimum energy. δ able scatter in t values. REFERENCES CONCLUSIONS 1. G. S. Konstantinova and V. N. Lozovskiœ, Kristal- lograﬁya 44, 698 (1999) [Crystallogr. Rep. 44, 649 The results obtained show that the appearance of (1999)]. grain boundaries forming a groove when intersecting the crystallization front during epitaxial growth is essen- 2. Ch. V. Kopetskiœ, A. N. Orlov, and L. K. Fionova, Grain Boundaries in Elemental Materials (Nauka, Moscow, tially dependent on the growth conditions, namely, on 1987). the angle formed by the boundary with the normal to the crystallization front and the existence of a tangen- 3. A. V. Andreeva and L. K. Fionova, Fiz. Met. Metalloved. tial temperature gradient. Diffusion ﬂows in the groove 52 (3), 593 (1981). volume arising due to the above factors lead to its displacement along the crystallization front. This process Translated by V. Semenov

CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002

INFORMATION

Report on the Results of the 2001 Struchkov Prize Competition for Young Scientists and Announcement of the 2002 Competition

The Struchkov Prize has been awarded annually amounted to 3000 roubles each. Any winner of the since 1997 for the best scientific study in crystal chem- incentive prize is allowed to participate in future com- istry and the application of X-ray structure analysis to petitions. the solution of chemical problems. Researchers who To take part in the 2002 competition, a competitor are residents of the Commonwealth of Independent should present to the Center of X-ray Structure Studies States or the Baltic states and are under 36 years of age no later than June 1, 2002 the following documents: at the time of the presentation of their documents are invited to participate in the competition. Each scientific 1. A completed competitor form including his/her: study is presented on behalf of a single author, and each (a) full name; author is allowed to present only one work a year. The results presented to the competition should be pub- (b) title of the scientific study presented to the com- lished or submitted for publication in a refereed jour- petition; nal. The winner of the competition is decided by a spe- (c) date of birth; cial competition jury that is formed by the leaders of the Center of X-ray Structure Studies and consists of lead- (d) scientific degree and post; ing Russian scientists in crystal chemistry and X-ray (e) affiliation; structure analysis. The decision of the jury will be announced not later than November 1 this year. (f) postal address of the institution; Sixteen young scientists from Moscow, Cher- (g) office telephone; nogolovka, Novosibirsk, Kazan, Tashkent, and Chi- (h) e-mail; sinau took part in the 2001 competition. (i) full name of the supervisor (for those who have In 2001, two Struchkov Prizes were awarded. The supervisors). winners are 2. An abstract of the study (prepared accurately and M.A. Zakharov, a postgraduate from the Depart- not exceeding three pages printed with a line spacing of ment of General Chemistry, Faculty of Chemistry, one and a half, 38 lines to a page, and 65 characters to Moscow State University, for his study entitled “Syn- a line) containing a clear explanation of the competi- thesis and Crystal Structure of Acid Selenates”; and tor’s contribution to the study. Ya.V. Zubavichus, a researcher from the Labora- 3. A list of papers published or submitted for publi- tory of the Structural Studies of Polymers, Nesmey- cation relating to the subject of the study presented to anov Institute of Organoelement Compounds, Russian the competition. Academy of Sciences, for his study entitled “Structural Characterization of Partially Ordered Layered Nano- 4. Reprints or photocopies of all or some of these composites Based on Molybdenum Disulfide.” papers (at the author’s discretion). The winners of the competition received diplomas The documents should be sent to the Center of and monetary awards. In addition, nine incentive prizes X-ray Structure Studies, Nesmeyanov Institute of were awarded to the following researchers: S.B. Ar- Organoelement Compounds, Russian Academy of Sci- temkina (Novosibirsk), A.V. Churakov (Moscow), ences, ul. Vavilova 28, Moscow, 117813 Russia. K.K. Turgunov (Tashkent), Yu.K. Gubina (Moscow), The data indicated in paragraphs 1Ð3 should be pre- E.V. Karpova (Moscow), Yu.V. Torubaev (Moscow), sented to the jury as MS DOS ASCII files. The files can I.S. Neretin (Moscow), O.N. Kazheva (Cher- be sent by post on a diskette or by e-mail: nogolovka), and O.A. Lodochnikova (Kazan). [email protected]. For further information, Since 2000, the competition has been sponsored by contact the Center by tel. (095) 135-9271 or e-mail: the Struchkov Prize Society, which is the International [email protected] or [email protected]. Crystallographic Association of former students and colleagues of Yu.T. Struchkov. The 2001 prizes amounted to 24000 roubles, and the incentive prizes Translated by I. Polyakova